and healed all who were sick, that it might be fulfilled which was spoken by Isaiah the prophet...
Peter is applying this to Jesus healing people while He was on earth physically present with them doing ministry, not now by the means of the Spirit...
You can't jump to that conclusion, this was before the cross...
Is. 53:4 Surely he hath borne our griefs, and carried our sorrows: yet we did esteem him stricken, smitten of God, and afflicted.
The Holy Spirit translates this verse very differently to our modern translators...
In Matthews gospel the Holy Spirit (through Matthew) translates this very differently...
Matt. 8:17 That it might be fulfilled which was spoken by Esaias the prophet, saying, Himself took our infirmities, and bare our sicknesses.
The Holy Spirit translates these words as infirmities and sicknesses and says that Jesus took them, some translations say he bore them...
So I went back to look at this word translated griefs in Isa. 53:4 and followed it all through the bible and it is very interesting what I found...
Chali Sickness or Disease?
Deu 7:15 - And the LORD will take away from thee all sickness[2483], and will put none of the evil diseases of Egypt, which thou knowest , upon thee; but will lay them upon all [them] that hate thee.
Deu 28:59 - Then the LORD will make thy plagues wonderful , and the plagues of thy seed, [even] great plagues, and of long continuance , and sore sicknesses[2483], and of long continuance .
Deu 28:61 - Also every sickness[2483], and every plague, which [is] not written in the book of this law, them will the LORD bring upon thee, until thou be destroyed .
1Ki 17:17 - And it came to pass after these things, [that] the son of the woman, the mistress of the house, fell sick ; and his sickness[2483] was so sore, that there was no breath left in him.
2Ki 1:2 - And Ahaziah fell down through a lattice in his upper chamber that [was] in Samaria, and was sick : and he sent messengers, and said unto them, Go , enquire of Baalzebub the god of Ekron whether I shall recover of this disease[2483].
2Ki 8:8 - And the king said unto Hazael, Take a present in thine hand, and go, meet the man of God, and enquire of the LORD by him, saying , Shall I recover of this disease[2483]?
2Ki 8:9 - So Hazael went to meet him, and took a present with him, even of every good thing of Damascus, forty camels' burden, and came and stood before him, and said , Thy son Benhadad king of Syria hath sent me to thee, saying , Shall I recover of this disease[2483]?
2Ki 13:14 - Now Elisha was fallen sick of his sickness[2483] whereof he died . And Joash the king of Israel came down unto him, and wept over his face, and said , O my father, my father, the chariot of Israel, and the horsemen thereof.
2Ch 16:12 - And Asa in the thirty and ninth year of his reign was diseased in his feet, until his disease[2483] [was] exceeding [great]: yet in his disease[2483] he sought not to the LORD, but to the physicians .
2Ch 21:15 - And thou [shalt have] great sickness[2483] by disease of thy bowels, until thy bowels fall out by reason of the sickness[2483] day by day.
2Ch 21:18 - And after all this the LORD smote him in his bowels with an incurable disease[2483].
2Ch 21:19 - And it came to pass, that in process of time, after the end of two years, his bowels fell out by reason of his sickness[2483]: so he died of sore diseases. And his people made no burning for him, like the burning of his fathers.
Psa 41:3 - The LORD will strengthen him upon the bed of languishing: thou wilt make all his bed in his sickness[2483].
Ecc 5:17 - All his days also he eateth in darkness, and [he hath] much sorrow and wrath with his sickness[2483].
Ecc 6:2 - A man to whom God hath given riches, wealth, and honour, so that he wanteth nothing for his soul of all that he desireth , yet God giveth him not power to eat thereof, but a stranger eateth it: this [is] vanity, and it [is] an evil disease[2483].
Isa 1:5 - Why should ye be stricken any more? ye will revolt more and more : the whole head is sick[2483], and the whole heart faint.
Isa 38:9 - The writing of Hezekiah king of Judah, when he had been sick , and was recovered of his sickness[2483]:
Isa 53:3 - He is despised and rejected of men; a man of sorrows, and acquainted with grief[2483]: and we hid as it were [our] faces from him; he was despised , and we esteemed him not.
Isa 53:4 - Surely he hath borne our griefs[2483], and carried our sorrows: yet we did esteem him stricken , smitten of God, and afflicted .
Jer 6:7 - As a fountain casteth out her waters, so she casteth out her wickedness: violence and spoil is heard in her; before me continually [is] grief[2483] and wounds.
Jer. 6:7 ESV As a well keeps its water fresh, so she keeps fresh her evil; violence and destruction are heard within her; sickness and wounds are ever before me.
Jer 10:19 - Woe is me for my hurt! my wound is grievous : but I said , Truly this [is] a grief[2483], and I must bear it.
Hsa 5:13 - When Ephraim saw his sickness[2483], and Judah [saw] his wound, then went Ephraim to the Assyrian, and sent to king Jareb : yet could he not heal you, nor cure you of your wound.
Almost without exception in other verses it is translated as either sickness or disease...
Why not in Isa 53:4?
I believe it was because of the theology of the translators, they didn't believe healing was in the atonement and so they used the word griefs instead, but thank God that the Holy Spirit revealed the truth to us in Matt. 8:17