Here is the Orthodox response to Papal infallibility and Papal supremacy:
From this encyclical we can see that it's going to take a miracle, because the Catholics have these two recently declared dogmas that have been developing since the use of the term "Supreme Pontiff" or "Pontifex Maximus" to refer to the Pope of Rome.
The Patriarchal Encyclical of 1895
A Reply to the Papal Encyclical of Pope Leo XIII (1895) on Reunion
To the most Sacred and Most Divinely-beloved Brethren in Christ the Metropolitans and Bishops, and their sacred and venerable Clergy, and all the godly and Orthodox Laity of the Most Holy Apostolic and Patriarchal Throne of Constantinople.
Remember them which have the rule over you, who have spoken unto you the word of God: whose faith follow, considering the end of their own conversation: Jesus Christ the same yesterday, and today, and for ever. Be not carried about with divers and strange doctrines. (Heb. xiii. 7, 8).
I
Every godly and orthodox soul, which has a sincere zeal for the glory of God, is deeply afflicted and weighed down with great pain upon seeing that he, who detests that which is good and is a murderer from the beginning, impelled by envy of man's salvation, never ceases continually to sow divers tares in the field of the Lord, in order to sift the wheat. From this source indeed, even from the earliest times, there sprang up in the Church of God heretical tares, which have in many ways made havoc, and do still make havoc, of the salvation of mankind by Christ; which moreover, as bad seeds and corrupted members, are rightly cut off from the sound body of the Orthodox Catholic Church of Christ. But in these last times the evil one has rent from the orthodox Church of Christ even whole nations in the West, having inflated the bishops of Rome with thoughts of excessive arrogance, which has given birth to divers lawless and anti-evangelical innovations. And not only so, but furthermore the Popes of Rome from time to time, pursuing absolutely and without examination modes of union according to their own fancy, strive by every means to reduce to their own errors the catholic Church of Christ, which throughout the world walks unshaken in the orthodoxy of faith transmitted to her by the Fathers.
II
Accordingly, the Most Blessed Pope of Rome, Leo XIII, on the occasion of his episcopal jubilee, published in the month of June of the year of grace 1895 an encyclical letter, addressed to the leaders and peoples of the world, by which he also at the same time invites our Orthodox Catholic and Apostolic Church of Christ to unite with the papal throne, thinking that such union can only be obtained by acknowledging him as supreme pontiff and the highest spiritual and temporal ruler of the universal Church, as the only representative of Christ upon earth and the dispenser of all grace.
III
No doubt every Christian heart ought to be filled with longing for union of the Churches, and especially the whole orthodox world, being inspired by a true spirit of piety, according to the divine purpose of the establishment of the church by the God-man our Saviour Christ, ardently longs for the unity of the Churches in the one rule of faith, and on the foundation of the apostolic doctrine handed down to us through the Fathers, Jesus Christ Himself being the chief corner stone.1 Wherefore she also every day, in her public prayers to the Lord, prays for the gathering together of the scattered and for the return of those who have gone astray to the right way of the truth, which alone leads to the Life of all, the only-begotten Son and Word of God, our Lord Jesus Christ.2 Agreeably, therefore, to this sacred longing, our orthodox Church of Christ is always ready to accept any proposal of union, if only the Bishop of Rome would shake off once for all the whole series of the many and divers anti-evangelical novelties that have been privily brought in to his Church, and have provoked the sad division of the Churches of the East and West, and would return to the basis of the seven holy Ecumenical Synods, which, having been assembled in the Holy Spirit, of representatives of all the holy Churches of God, for the determination of the right teaching of the faith against heretics, have a universal and perpetual supremacy in the Church of Christ. And this, both by her writings and encyclical letters, the Orthodox Church has never ceased to intimate to the Papal Church, having clearly and explicitly set forth that so long as the latter perseveres in her innovations, and the orthodox Church adheres to the divine and apostolic traditions of Christianity, during which the Western Churches were of the same mind and were united with the Churches of the East, so long is it a vain and empty thing to talk of union. For which cause we have remained silent until now, and have declined to take into consideration the papal encyclical in question, esteeming it unprofitable to speak to the ears of those who do not hear. Since, however, from a certain period the Papal Church, having abandoned the method of persuasion and discussion, began, to our general astonishment and perplexity, to lay traps for the conscience of the more simple orthodox Christians by means of deceitful workers transformed into apostles of Christ,3 sending into the East clerics with the dress and headcovering of orthodox priests, inventing also divers and other artful means to obtain her proselytizing objects; for this reason, as in sacred duty bound, we issue this patriarchal and synodical encyclical, for a safeguard of the orthodox faith and piety, knowing that the observance of the true canons is a duty for every good man, and much more for those who have been thought worthy by Providence to direct the affairs of others.4
IV
The union of the separated Churches with herself in one rule of faith is, as has been said before, a sacred and inward desire of the holy, catholic and orthodox apostolic Church of Christ; but without such unity in the faith, the desired union of the Churches becomes impossible. This being the case, we wonder in truth how the Most Blessed Pope Leo XIII, though he himself also acknowledges this truth, falls into a plain self-contradiction, declaring, on the one hand, that true union lies in the unity of faith, and, on the other hand, that every Church, even after the union, can hold her own dogmatic and canonical definitions, even when they differ from those of the Papal Church, as His Blessedness declares in a previous encyclical, dated November 30, 1894. For there is an evident contradiction when in one and the same Church one believes that the Holy Spirit proceeds from the Father, and another that He proceeds from the Father and the Son; when one sprinkles, and another baptizes (immerses) thrice in the water; one uses leavened bread in the sacrament of the Holy Eucharist, and another unleavened; one imparts to the people of the chalice as well as of the bread, and the other only of the holy bread; and other things like these. But what this contradiction signifies, whether respect for the evangelical truths of the holy Church of Christ and an indirect concession and acknowledgement of them, or something else, we cannot say.
V
But however that may be, for the practical realization of the pious longing for the union of the Churches, a common principle and basis must be settled first of all; and there can be no such safe common principle and basis other than the teaching of the Gospel and of the seven holy Ecumenical Councils. Reverting, then, to that teaching which was common to the Churches of the East and of the West until the separation, we ought, with a sincere desire to know the truth, to search what the one holy, catholic and orthodox apostolic Church of Christ, being then of the same body, throughout the East and West believed, and to hold this fact, entire, and unaltered. But whatsoever has in later times been added or taken away, every one has a sacred and indispensable duty, if he sincerely seeks for the glory of God more than for his own glory, that in a spirit of piety he should correct it, considering that by arrogantly continuing in the perversion of the truth he is liable to a heavy account before the impartial judgement-seat of Christ. In saying this we do not at all refer to the differences regarding the ritual of the sacred services and the hymns, or the sacred vestments, and the like, which matters, even though they still vary, as they did of old, do not in the least injure the substance and unity of the faith; but we refer to those essential differences which have reference to the divinely transmitted doctrines of the faith, and the divinely instituted canonical constitution of the administration of the Churches. In cases where the thing disregarded is not the faith (says also the holy Photius),5 and is no falling away from any general and catholic decree, different rites and customs being observed among different people, a man who knows how to judge rightly would decide that neither do those who observe them act wrongly, nor do those who have not received them break the law.6
VI
And indeed for the holy purpose of union, the Eastern orthodox and catholic Church of Christ is heartily ready to accept all that which both the Eastern and Western Churches unanimously professed before the ninth century, if she has perchance perverted or does not hold it. And if the Westerns prove from the teaching of the holy Fathers and the divinely assembled Ecumenical Councils that the then orthodox Roman Church, which was throughout the West, even before the ninth century read the Creed with the addition, or used unleavened bread, or accepted the doctrine of a purgatorial fire, or sprinkling instead of baptism, or the immaculate conception of the ever-Virgin, or the temporal power, or the infallibility and absolutism of the Bishop of Rome, we have no more to say. But if, on the contrary, it is plainly demonstrated, as those of the Latins themselves, who love the truth, also acknowledge, that the Eastern and orthodox catholic Church of Christ holds fast the anciently transmitted doctrines which were at that time professed in common both in the East and the West, and that the Western Church perverted them by divers innovations, then it is clear, even to children, that the more natural way to union is the return of the Western Church to the ancient doctrinal and administrative condition of things; for the faith does not change in any way with time or circumstances, but remains the same always and everywhere, for there is one body and one Spirit, it is said, even as ye are called in one hope of your calling; one Lord, one faith, one baptism, one God and Father of all, who is above all, and through all, and in you all.7
VII
So then the one, holy, catholic, and apostolic Church of the seven Ecumenical Councils believed and taught in accordance with the words of the Gospel, that the Holy Spirit proceeds from the Father; but in the West, even from the ninth century, the holy Symbol of Faith, which was composed and sanctioned by Ecumenical Councils, began to be falsified, and the idea that the Holy Spirit proceeds also from the Son to be arbitrarily promulgated. And certainly Pope Leo XIII is not ignorant that his orthodox predecessor and namesake, the defender of orthodoxy, Leo III, in the year 809 denounced synodically this anti-evangelical and utterly lawless addition, and from the Son (Filioque); and engraved on two silver plates, in Greek and Latin, the holy Symbol of Faith of the first and second Ecumenical Synods, entire and without any addition; having written moreover, These words I, Leo, have set down for love and as a safeguard of the orthodox faith (Haec Leo posui amore et cautela fidei orthodoxa).8
Likewise he is by no means ignorant that during the tenth century, or at the beginning of the eleventh, this anti-evangelical and lawless addition was with difficulty inserted officially into the holy Symbol of Faith at Rome also, and that consequently the Roman Church, in insisting on her innovations, and not coming back to the dogma of the Ecumenical Synods, renders herself fully responsible before the one holy, catholic, and apostolic Church of Christ, which holds fast that which has been received from the Fathers, and keeps the deposit of the faith which was delivered to it unadulterated in all things, in obedience to the Apostolic injunction: That good thing which was committed unto thee keep by the Holy Spirit which dwelleth in us; avoiding profane and vain babblings, and oppositions of science falsely so called: which some professing have erred concerning the faith.9
VIII
VIII. The one holy, catholic and apostolic Church of the first seven Ecumenical Synods baptized by three immersions in the water, and the Pope Pelagius speaks of the triple immersion as a command of the Lord, and in the thirteenth century baptism by immersions still prevailed in the West; and the sacred fonts themselves, preserved in the more ancient churches in Italy, are eloquent witnesses on this point; but in later times sprinkling or effusion, being privily brought in, came to be accepted by the Papal Church, which still holds fast the innovation, thus also widening the gulf which she has opened; but we Orthodox, remaining faithful to the apostolic tradition and the practice of the seven Ecumenical Synods, stand fast, contending for the common profession, the paternal treasure of the sound faith.10
IX
The one holy, catholic and apostolic Church of the seven Ecumenical Synods, according to the example of our Saviour, celebrated the divine Eucharist for more than a thousand years throughout the East and West with leavened bread, as the truth-loving papal theologians themselves also bear witness; but the Papal Church from the eleventh century made an innovation also in the sacrament of the divine Eucharist by introducing unleavened bread.
X
The one holy, catholic and apostolic Church of the seven Ecumenical Synods held that the precious gifts are consecrated after the prayer of the invocation of the Holy Spirit by the blessing of the priest, as the ancient rituals of Rome and Gaul testify; nevertheless afterwards the Papal Church made an innovation in this also, by arbitrarily accepting the consecration of the precious gifts as taking place along with the utterance of the Lord's words: Take, eat; this is my body: and Drink ye all of it; for this is my blood.11
to be continued