Peter DID do what he was accused of. He drew back from eating with the Gentiles because he feared the circumcision party when they rolled into town. This was not in line with the truth of the gospel in that there is no longer any dividing wall of hostility between Jew and Gentile because God in Christ made one new man out of the two.
Paul rebuked Peter publicly because it was that serious, Matthew 18 does not apply in this case. We’re not dealing with a private wrong but a public error by an apostle. Clearly God used Paul here to correct Peter. My sense is that Peter received it as from God.
In his defense of his God-given authority, Paul builds a case in which this event is one of the proofs. For some reason you don’t accept that God shows no partiality and WILL rebuke an apostle who seemed to be a pillar to many, but is really just a man needling correction from time to time as we all do.
Peter DID do what he was accused of.
You say that Peter did what he was accused, how do you know?
You say Mathew 18 dose not apply in this case, oh yes it does.
Mathew 18
15 Moreover if thy brother shall trespass against thee, go and tell him his fault between thee and him alone: if he shall hear thee, thou hast gained thy brother.
16 But if he will not hear thee, then take with thee one or two more, that in the mouth of two or three witnesses every word may be established.
17 And if he shall neglect to hear them, tell it unto the church: but if he neglect to hear the church, let him be unto thee as an heathen man and a publican.
18 Verily I say unto you, Whatsoever ye shall bind on earth shall be bound in heaven: and whatsoever ye shall loose on earth shall be loosed in heaven.
If you read it closely you can see that Jesus is talking about a public matter, the church, if it was a private matter why would the Church be involved.
Paul accused Peter of leaving the gentiles and eating with the Jews, it was nothing more than an accusation, whether true or not I have no idea but I doubt any one who will spread talk about some one.