Galatians chapter 1 is a declaration supporting the authenticity of Paul as a servant of Christ (verse 10) sent by God to preach Christ to the Gentiles (verse 16). Three years after Paul’s dramatic conversion, he went to Jerusalem and stayed with Peter for fifteen days (verse18). Fourteen years later Paul returned to Jerusalem to speak with “those who seemed to be leaders” about his revelation to preach to the Gentiles (Chapter 2, verse 2).
Galatians 2:7-9 shows that Paul recognised Peter as an apostle and a servant of God:
For God, who was at work in the ministry of Peter as an apostle to the Jews, was also at work in my ministry as an apostle to the Gentiles. James, Peter and John, those reputed to be pillars, gave me and Barnabas the right hand of fellowship when they recognised the grace given to me.
However, he nowhere suggests that Peter has the highest authority within the church. Later, when Peter came to Antioch, there was a serious difference of opinion between Peter and Paul. Had Paul recognised Peter as the earthly leader of the Church it seems unlikely that he would have opposed him to his face (verse 11). But no, he accused Peter of hypocrisy.
The conclusion of the matter is that Paul did not give in to Peter. In Galatians 6:12 Paul condemns the Judaisers of trying to make a good impression outwardly in order to avoid being persecuted. The only authority Paul recognised and submitted to was the authority of Christ Jesus, who is the one and the only head of his church:
And he is the head of the body, the church; he is the beginning and the firstborn from among the dead, so that in everything he might have the supremacy.