This is categorically untrue; I have discussed the actual history of the early church and Emperor Constantine’s involvement several times and am disinclined to repeat myself, but suffice it to say, Constantine’s son Constantius became an Arian, and the early church, founded by the Holy Apostles including Saints Peter, Paul and Thomas (who collectively established the largest number of permanent churches), along with Saints John, James the Just, Andrew, and the other Apostles, was persecuted by the non-Trinitarian Arian church from roughly the time St. Athanasius was exiled in 336 AD (when St. Constantine was still on the throne, but his pious mother St. Helena had reposed, and he was likely senile, and was being manipulated by the sinister bishop Eusebius of Nicomedia), until at least 386 AD. And then, while the dogmatic persecution stopped, we had the grotesque spectacle of Eudoxia conspiring with Theophilus of Alexandria to depose St. John Chrysostom after he criticized her for purchasing a solid gold toilet (or the fourth century equivalent thereof…the Romans did have plumbing).
Physical persecution resumed in the 6th century with the genocidal persecution of the Oriental Orthodox under Emperor Justinian, then in the 7th century the Emperor of the time decided it would be a good idea to cut off the tongue of St. Maximus the Confessor for his opposing the Monothelite heresy (which had been supported decades earlier by Pope Honorius I of Rome; it was around the time of St. Maximus declared anathema by the Sixth Ecumenical Council). And then the Empire decided to enforce iconoclasm, and horribly persecuted Iconodules, despite the ruling of the Second Council of Nicaea in 787 AD that Iconoclasm was forbidden. This lasted until 843 AD. Only after this time did the adversarial relationship between the Church and the Byzantine Empire cease, as instead a new adversarial relationship between the Carolignian Holy Roman Empire and the Roman church, and between the Roman church and the Eastern Orthodox church, and between the Byzantine Empire and Venice in the West and the Muslims in the East, both eager to divide up the remaining Byzantine lands. Spoiler: the Muslims won, although their path to victory was greatly aided by the Fourth Crusade, in which Venice under the pretense of liberating the Holy Land once more instead just conquered Constantinople.