No, that's incorrect. There is one gospel and both men preached the same gospel. Peter had an emphasis for most of his ministry to go to Jewish people and Paul went to Gentiles, but they didn't preach different ways to get saved. You can read Peter's sermons in the book of Acts and see that. Paul felt strongly enough about that that he said that if someone preached a message other than the one he preached they should be accuser. Was Paul talking about Peter? No, of course not.
8 "But even if we or an angel from heaven should preach a gospel other than the one we preached to you, let them be under God’s curse! 9 As we have already said, so now I say again: If anybody is preaching to you a gospel other than what you accepted, let them be under God’s curse!" Galatians 1:8-9, NIV
Peter and Paul did not preach the same gospel messages. Below are the differences between the gospels and Apostleship's of Peter and Paul.
Peter's Apostleship: His Apostleship was commanded by Jesus Christ along with the other eleven disciples before Paul. Peter was primarily the Apostle to Israel (Galatians 2:8). Peter's Apostleship was diminishing after Paul was converted.
Paul's Apostleship: Paul was persecuting the church of God before being converted. After his conversion Paul was given revelations from the Lord Jesus Christ; Paul did not learn from any man including the twelve Apostles (Galatians 1:12). Paul was primarily the Apostle to Gentiles (Romans 11:13 2 Timothy 1:11).
Peter's Justification: Peter and the other eleven Apostles taught justification unto eternal life by believing on the name of Jesus; by believing Jesus is Christ, the Son of God (Acts 10:43 Matthew 16:16 Acts 4:12 Acts 2:36 John 20:31).
Peter preached that Israel received the new covenant by the crucifixion and shed blood of Jesus (Matthew 26:28 1 Peter 1:2 1 Peter 1:19 Acts 3:18 1 Peter 2:24 Acts 3:25). Peter preached forgiveness of sin at the coming of the Lord Jesus Christ (Acts 3:19).
Paul's Justification: Paul taught justification unto eternal life by the cross (1 Corinthians 1:18 Galatians 6:14 Colossians 1:20). Justification by faith that Christ died for our sins, was buried and rose again for complete salvation unto eternal life (1 Corinthians 15:1-4 Colossians 2:10). Paul taught forgiveness of all sin now by the blood of Jesus (Ephesians 1:7).
Peter's gospel: Peter preached remission of sins in the name of Jesus (Acts 10:43). Peter taught the gospel of the kingdom (Luke 9:2 Luke 9:6 2 Peter 1:11). Israel would inherit the promise of the Davidic kingdom on earth (Luke 1:32 Genesis 13:15). Israel would rule and reign with Jesus for a thousand year period (Revelation 20:6). Israel would be a light for salvation unto the Gentile nations that would enter the kingdom (Isaiah 42:6).
Paul's gospel: Paul taught the gospel of the uncircumcision (Galatians 2:7). Paul taught salvation to all freely by faith in Christ apart from the covenants of Israel and apart from keeping the law (Romans 9:4 Romans 3:28 Galatians 2:19).
Peter and the resurrection: Peter preached the gospel of the circumcision (Galatians 2:7). Israel under the old covenant promises was promised to be a holy nation and a kingdom of kings and priests to rule over the Gentile nations with Christ (Revelation 1:6 Exodus 19:6 Revelation 5:10). Peter preached that the resurrection of Christ gave Israel assurance of that coming kingdom because Christ was raised to sit on the throne of David in that prophetic kingdom (1 Peter 1:3 Acts 3:25 Acts 2:30). Peter preached the receiving of the Holy Ghost to enable Israel to enter the kingdom (Jeremiah 31:33 Acts 2:33 Acts 2:38).
Paul and the resurrection: Paul taught that Christ was risen for our justification (Romans 4:25).
Peter and works: Peter preached works as necessary for salvation in order to enter the kingdom (Acts 10:35 1 Peter 1:15 1 Peter 2:9 2 Peter 1:10-11). Peter preached to Israel repentance and baptism for the remission of sins (Acts 2:38). Peter preached holiness and to remain steadfast unto the coming of Jesus and his kingdom (2 Peter 3:17).
Paul and works: Paul taught salvation without works (Titus 3:5 Romans 4:6 Ephesians 2:8-9).
Peter and the church: Peter preached to Israel (Acts 3:12). Gentiles had to come through Israel and it's covenants to be added to the church (Exodus 12:48 Acts 11:1); Gentiles had to believe the preaching of the word delivered to Israel as preached by Peter (Acts 13:48).
Paul and the church: Paul taught the church as the body of Christ, a new creature where there was neither Jew or Gentile (Galatians 6:15 Galatians 3:28 Ephesians 2:14).
Peter and heaven: Peter taught that Israel would receive the kingdom on earth ( Luke 9:2 Luke 9:6 2 Peter 1:11 Acts 2:30). He taught their reward was stored in heaven but not in heaven (1 Peter 1:4).
Paul and heaven: Paul taught the church seated with Christ in heavenly places (Ephesians 2:6).
Both Peter and Paul were ordained of God to preach the gospel. Those who believed their gospels are in Christ. Peter's gospel faded away as Israel was diminished and is not the gospel that is preached today. Paul preaches the gospel by which we are saved today. Paul in his epistles reveal the mysteries and doctrine for the church today.
