Well, we know that Peter travelled to Antioch (in about 51AD or so), wherein Galatians 2:11-21 relates the record of Paul's rebuke of Peter.
It is occasionally argued by some Protestants (and some Eastern Orthodox) that Paul's rebuke of Peter constitutes evidence against the doctrine of Papal Supremacy and Infallibility -- but it doesn't (this is really a poor logical argument). Strictly speaking, neither hypocrisy or cowardice on Peter's part nor the rebuke thereof by a fellow Apostle (Paul), would itself comprise particularly strong evidence against Papal Supremacy and Infallibility (assuming such a dogma for the sake of discussion).
The real argument against Papal Supremacy and Infallibility found in Galatians 2 is much stronger when we consider just WHY Peter fell into hypocrisy concerning the Gentile Christians -- Galatians 2:12 explicitly tells us that it was because Peter feared James.
Now, if Peter was supposedly "Pope" -- then Peter's fear of James is quite an awkward square peg for one to fit into the hollow circle of Papal Supremacy. Instead, what we see in Scripture is the administrative supremacy of James the Righteous, the half-brother (or, according to the Eastern Orthodox view, the step-brother) of Jesus Christ and Bishop of Jerusalem. Peter reported to James (Acts 12:17); It was James, not Peter, who passed Judgment at the Jerusalem Council (Acts 15:13-22); James is accounted as the foremost of the Bishops (Acts 21:18); and as noted above, Peter himself feared the authority of James (Galatians 2:12).
A Jerusalem-centered Early Church under the general administration of James the Righteous thus accords much better with the actual record of Acts than any supposed Roman Papacy -- and better explains the Galatians 2 happenings in Antioch, as well. I think Xzins might have some thoughts on the matter (if he wants to add any comments).
Does that pretty well cover it, or had you further questions? Best, OP
This is not a 'real argument' at all, it's a flimsy personal intrepretation that falls like a stone to the ground in the face of correction. Here are the verses you claim prove that Peter was not head of the Apostles:
"But when Cephas came to Antioch I withstood him to his face, because he was to be blamed. For before some men came from James, he did eat with the Gentiles; but when they came he drew back and separated himself, fearing them who were of the circumcision. And to his dissimulation the rest of the Jews consented, so that Barnabus was also led by them into that dissimulation. But when I saw that THEY were not walking uprightly unto the truth of the gospel, I said to Cephas before them all: If thou, being a Jew, live after the manner of the Gentiles, and not as the Jews do, how doest thou compel the Gentiles to live as the Jews?" (2 Gal. 11-15)
You claim that Peter "feared" James, but the Bible says that he feared the entire "circumsion party". The fear spoken of here regarding Peter, Barnabus, and the entire party, probably refers to their human pride. It could easily have been that Peter and the rest didn't want the righteous 'circumcision party' from James to see them "mingling with the rabble", as it were.
We know from Peter's three denials of Christ, his momentary loss of faith and falling into the sea, and his taking up his sword at Gethsemene that he was prone to fear. All this means is that he was human, but his fear was conquered when he kept his faith to the end and was martyred for it at Rome.
But far more importantly, the verses tell us that when Peter acted insincerely the entire Jewish party, including Barnabus himself, then followed his error. Unless Peter were clearly the Head, they would not have followed him; which also rasies the question "why would Paul choose to rebuke only Peter even though the entire Jewish party, (surely consisting of elders and leaders), including Barbabus himself, all fell into the same error"? Paul didn't correct them all because he corrected the entire Church by correcting Peter, the head of the Church.
The Catholic Church is rife with examples of the Saints 'rebuking' (correcting) Popes in non-doctrinal matters. Your example of Peter being rebuked is not even the first we know of, for Jesus had to correct Peter several times when He still walked amongst the Apostles. I will wager that even Peter's mother-in-law corrected him a few times, :o)
Nobody ever claimed that the Popes were perfect, we only claim that that when they teach matters of Christian faith and morals, (matters directly affecting salvation), to the Church that their teachings cannot err.