Why did Peter say the lame man was healed through Jesus Christ 'whom you crucified'?
Peter proclaimed the healing came through Jesus Christ of Nazareth, whom Israel had crucified, to confront the nation with its awful sin of rejecting their Messiah. He used the miracle as a call to national repentance, showing their guilt and God's continued offer of mercy if they turned back to Him.
Les explains that Peter and John were dealing with the lame man, and the nation of Israel was upset about how it happened. Peter tried to explain that it was through the power of Jesus Christ of Nazareth whom they had crucified. Back in Acts 3:19-21, Peter said if the nation would repent—primarily of the awful sin of having rejected their Messiah, as well as all their other national sins—then God would respond. Les connects this call to repentance with prophecy and Israel's need to turn back to God. Thus Peter's mention of “whom you crucified” pressed Israel's guilt and the need for repentance.