The general reading of Gal 2:11-12 etc… would imply that Paul felt Peter was being disrespectful in the way he was treating the gentiles and treating the Jews more favourably and that all should be treated equally.
Unfortunately, we only have Pauls side of the story and not Peters or anyone else’s.
Why would they be arguing – did Peter or James not know what Jesus was preaching, was Jesus not following Abraham and Moses and David lineage?
However, looking at other supporting passages, there appears to be a conflict between Peter and Paul to become the leader of the Gentiles
Paul carries on arguing against the others Gal 2:11-21 – basically condemning all Jewish Christians including Peter, James & Barnabas.
Gal 2:13The other Jews joined him in his hypocrisy, so that by their hypocrisy even Barnabas was led astray 14When I saw that they were not acting in line with the truth of the gospel, I said to Peter in front of them all, "You are a Jew, yet you live like a Gentile and not like a Jew. How is it, then, that you force Gentiles to follow Jewish customs?
Some examples of Paul breaking the Law
Gal: 2:15"We who are Jews by birth and not 'Gentile sinners' 16know that a man is not justified by observing the law, but by faith in Jesus Christ. So we, too, have put our faith in Christ Jesus that we may be justified by faith in Christ and not by observing the law, because by observing the law no one will be justified.
Romans 14:20 - Do not destroy the work of God for the sake of food. All food is clean, but it is wrong for a man to eat anything that causes someone else to stumble.
Galatians 3:1-3 - 1You foolish Galatians! Who has bewitched you? Before your very eyes Jesus Christ was clearly portrayed as crucified. 2I would like to learn just one thing from you: Did you receive the Spirit by observing the law, or by believing what you heard? 3Are you so foolish?
Genesis 17:14, the covenant is broken if there is no circumcision
14 But if any male is not circumcised, he will be cut off from his people; he has broken My covenant.”
Acts 15:29 - 29You are to abstain from food sacrificed to idols, from blood, from the meat of strangled animals and from sexual immorality. You will do well to avoid these things. Farewell.
Note 1 - Jesus followed the law below for more.
Paul will say and do anything to gain favour
1 Corinthians 9:21-22 - 21 When I am with the Gentiles who do not follow the Jewish law, I too live apart from that law so I can bring them to Christ. But I do not ignore the law of God; I obey the law of Christ.
22To the weak I became weak, to win the weak. I have become all things to all people so that by all possible means I might save some.
Paul is inventing his own Gospel
1 Corinthians 3:10 - According to the grace of God which is given unto me, as a wise masterbuilder, I have laid the foundation, and another buildeth thereon.
2 Timothy 2:8 - Remember that Jesus Christ of the seed of David was raised from the dead according
to my gospel:
2 Timothy 1:11 And of this gospel I was appointed a herald and an apostle and a teacher.
***Jesus says only one teacher - Matthew 23:8–12 8 But you are not to be called rabbi, for you have one teacher etc….
Peters right of authority & filled with the Holy Spirit
Acts 4:31 - After they prayed, the place where they were meeting was shaken. And they were all filled with the Holy Spirit and spoke the word of God boldly.
Matthew 16:18 - And I tell you that you are Peter, and on this rock I will build my church, and the gates of Hades will not overcome it.
Mark 3:14 – Then he appointed twelve of them and called them his apostles. They were to accompany him, and he would send them out to preach,
Rev 21:14 - 14 The wall of the city had twelve foundations bearing the names of the twelve apostles of the Lamb.
Act 15:7, 14 - 7 After much discussion, Peter got up and said to them, “Brothers, you know that in the early days God made a choice among you that the Gentiles would hear from my lips the message of the gospel and believe.
14 Peter has told you about the time God first visited the Gentiles to take from them a people for himself.
Paul does what he is told without question
Already suspicion Paul is not following the law and his preaching is hard to understand Note 2
2 Peter 3:15-16 - 16 He writes the same way in all his letters, speaking in them of these matters. His letters contain some things that are hard to understand, which ignorant and unstable people distort, as they do the other Scriptures, to their own destruction.
Acts 21:21-24 21 They have been informed that you teach all the Jews who live among the Gentiles to turn away from Moses, telling them not to circumcise their children or live according to our customs. 22 What shall we do? They will certainly hear that you have come, 23 so do what we tell you. There are four men with us who have made a vow. 24 Take these men, join in their purification rites and pay their expenses, so that they can have their heads shaved. Then everyone will know there is no truth in these reports about you, but that you yourself are living in obedience to the law.
Acts 15:22 - Then the apostles and elders, with the whole church, decided to select men from among them to send to Antioch with Paul and Barnabas. They chose Judas called Barsabbas and Silas, two leaders among the brothers,
Conclusion:
Clearly Peter has higher authority – Paul said nothing when told what to do & at best he was low level in the hierarchy that had to be accompanied by others.
Therefore, Paul had not authority to question or tarnish Peter. As I said at the beginning – we only have Paul’s side of the story. I’m sure Peters version would have been a lot different.
Note 1 - https://hermeneutics.stackexchange.com/a/60694/33268
Note 2 - https://hermeneutics.stackexchange.com/a/62976/33268