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