It proves the first Gentiles were saved by Peter's gospel, and it proves it is the same gospel that Paul preaches.
The message that saves that Peter was telling them is that:
1. God does not show favoritism.
That is the exact thing Paul says in Colossians 3:24, and Romans 2:11.
2. The good news of peace through Jesus Christ,
It is the same good news that Paul preached:
Romans 10:15 And how can anyone preach unless they are sent? As it is written: “How beautiful are the feet of those who bring good news!”
Romans 10:16But not all the Israelites accepted the good news. For Isaiah says, “Lord, who has believed our message?”
Did you read how Paul says not all the Israelites accepted the good news?
3. who is Lord of all.
Just like Paul says:
Romans 10:12For there is no difference between Jew and Gentile—the same Lord is Lord of all and richly blesses all who call on him,
4. He commanded us to preach to the people and to testify that he is the one whom God appointed as judge of the living and the dead.
It is what Paul taught:
2 Timothy 4:1 I charge you in the presence of God and of Christ Jesus, who will judge the living and the dead, and in view of His appearing and His kingdom:
5. All the prophets testify about him that everyone who believes in him receives forgiveness of sins through his name.”
That is what Paul taught:
Acts 26:20 First to those inDamascus and Jerusalem, then to everyone in theregion of Judea, and then to the Gentiles, I declared that they should repent and turn to God,performing deeds worthy of their repentance.
Acts 26:23 23that Christ would suffer, and as the first to rise from the dead, He would proclaim light to our people and to theGentiles.”