Jesus was born in December.

We know this because, when he was conceived in Mary's womb, Elizabeth was six months pregnant with John the Baptist.

John's birth was announced to Zachariah when he went to burn incense in the Temple. Zachariah's course of priestly service fell around the feast of Atonement, which falls around October. Counting six months forward from that places the conception of Christ near Passover, in March, and his birth in December.

This March date for the conception of Jesus agrees with many ancient traditions. The Jews believed the world was created in the month of Nisan on the Jewish calendar, which corresponds to March. Thus creation was renewed in Nisan by Christ's incarnation. It was also believed that a man was conceived and died on the same date. Given that Jesus died on the eve of Passover, this would place his conception likewise around Passover, in the month of Nisan/March. This dating agrees with ancient Christian sources, including the early Church Fathers to within two centuries of the time of Christ. This dating was not challenged until relatively recently.

Thus, the combined weight of Scriptural evidence and ancient Christian tradition supports the celebration of Christ's birth on December 25th, and it has nothing to do with any Pagan holidays.