Jesus instructs his disciples to handle personal disputes in this fashion: Moreover if thy brother shall trespass against thee, go and tell him his fault between thee and him alone: if he shall hear thee, thou hast gained thy brother. But if he will not hear thee, then take with thee one or two more, that in the mouth of two or three witnesses every word may be established. And if he shall neglect to hear them, tell it unto the church: but if he neglect to hear the church, let him be unto thee as an heathen man and a publican.Matthew 18:15-17
Why did Paul rebuke Peter in front of everyone and then write a letter recording this instance publicly? But when I saw that they walked not uprightly according to the truth of the gospel, I said unto Peter before them all, If thou…Galatians 2:14
Paul forces us to conclude he was following his own commandments, instead of Jesus's. Paul writes to Timothy, Them that sin rebuke before all, that others also may fear.
Value of this Evidence
This is fantastic evidence showing Paul did not follow the Way Jesus set out. If he had, Paul would have brought up his issues privately with Peter and company. Instead, Paul spread about the issue as much as he could, making himself a gossiper.