Peer was confident that Cornelius would know about the preaching of John the Baptist, the powerful signs performed by Jesus, Jesus’ death on a tree, God’s raising him on the third day and Jesus’ appearing to witnesses such as Peter.
Peter is acting here exactly as Isaiah’s servant of the Lord, Israel. Peter was blind spiritually but Jesus as the servant of the Lord (Israel-reduced-to-one) had healed Peter’s blindness through means of the Spirit whereby Peter had come to see the salvation act of God (Jesus’ death and resurrection). Now as part of Israel (the servant of the Lord healed-of-blindness) Peter proclaims that salvation act of God to the nations.
In addition to the above content, Peter also proclaims that Jesus is “the one appointed by God to be judge of the living and the dead. To him all the prophets bear witness that everyone who believes in him receives forgiveness of sins through his name.”