Why was Peter still appealing to Israel on the basis of Old Testament prophecy in Acts 3?
Peter still appealed to Israel on the basis of Old Testament prophecy because he did not yet understand the Church Age. He was still sticking to the Old Testament prophets and the Old Covenant program, expecting Israel to accept Jesus as Messiah so God could send Him as promised.
Les explains that Peter was still appealing to the nation on the basis of Old Testament prophecy because, as yet, Peter did not understand the Church Age—the calling out of the Body, a people for His name. Peter was still sticking to the Old Testament prophets and the Old Covenant program. He expected that after Israel rejected Him, the tribulation would unroll and then God would send Jesus, but that was dependent on Israel accepting the fact that He was their Messiah. Israel, however, continued to reject. The first part of Acts is nothing but Peter and the nation of Israel. Later, with the conversion of Saul and Peter's visit to Cornelius in chapter 10, a departure occurs: Israel goes off the scene, and Gentiles come into God's favor under the whole system of grace. In Acts 3, Peter even quotes Moses, showing he is still operating under the Old Covenant.