The stories of the incarnation do not prove that Jesus was the Son of God.  Some Liberal Christians would agree and would point out that although the Gospels of Matthew and Luke go to great lengths to show that Jesus’ birth fulfilled Messianic Prophecies from the Old Testament – including that the Messiah would be born of a virgin (Isaiah 7:14) and would come from Bethlehem (Micah 5:2) – they do not agree on the details and this suggests that the nativity stories could have developed in response to the demand for information about Jesus’ background rather than being historical records. John’s account of the Incarnation from Chapter 1:1-18 is even more different – although it is unequivocal in claiming that Jesus is the Son or “Word” of God, begotten not made, John does not mention Mary or Bethlehem at all.  John just asserts Jesus identity as a matter of faith rather than providing any proof.  Finally, scientists and scholars might point out that both stories about Jesus birth come from writers who are committed believers in Jesus as the Son of God, so they are likely to be biased and provide no credible, neutral evidence. 

Clearly, most Christians would disagree and argue that there are many similarities between Matthew and Luke or at least that their stories do not contradict each other.  They would say that it is very unusual to have so much information about Jesus’ birth after 2000 years from whatever source, and this suggests that Jesus’ was no ordinary baby and that his birth must have been remarkable enough to have been recorded or remembered in detail at the time. Further, they would point to the evidence of Jesus’ baptism (where God spoke to acknowledge Jesus as His Son), to the miracles and the Resurrection to support their belief that Jesus was God incarnate.  No ordinary human being nor even a prophet could have done what Jesus did.

Nevertheless, Christians would struggle to defend the claim that the stories of the incarnation (or any other part of the Gospels) PROVE Jesus’ identity as the son of God. In the end, stories are only stories, whether they are in the Bible or not.  No one of us knew Jesus or was in a position to ask or test what he meant when he called on God as His Father.  Whether anyone accepts that Jesus was God incarnate can only ever be a matter of faith.