Only Jesus Christ gave either man any greatness we may attribute to them.
And the reason this is an issue is because some one feels they have to put peter down by saying Paul had to correct peter.
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.