According to Taylor Marshall, it’s the 25th of December, and there’s Scriptural authority for it, based on the age of John the Baptist:
The second-century Protoevangelium of James also confirms a late September conception of the Baptist since the work depicts Saint Zacharias as High Priest and as entering the Holy of Holies — not merely the holy place with the altar of incense. This is a factual mistake because Zecharias was not the high priest, but one of the chief priests. Still, the Protoevangelium regards Zecharias as a high priest and this associates him with the Day of Atonement, which lands on the tenth day of the Hebrew month of Tishri (roughly the end of our September). Immediately after this entry into the temple and message of the angel Gabriel, Zacharias and Elizabeth conceive John the Baptist. Allowing for forty weeks of gestation, this places the birth of John the Baptist at the end of June — once again corresponding to the Catholic date for the Nativity of Saint John the Baptist on June 24.
The rest of the dating is rather simple. We read that just after the Immaculate Virgin Mary conceived Christ, she went to visit her cousin Elizabeth who was six months pregnant with John the Baptist. This means that John the Baptist was six months older that our Lord Jesus Christ (Luke 1:24-27, 36). Add six months to June 24 and it reveals December 24-25 as the birthday of Christ. Subtract nine months from December 25 and it reveals that the annunciation was March 25. All the dates match up perfectly.
So then, if John the Baptist was conceived shortly after the Jewish Day of the Atonement, then the traditional Catholic dates are essentially correct. The birth of Christ would be about or on December 25.
Of course, I am of the school of thought that believes Christmas should be moved to July, when the stores aren’t so crowded.
That said, I am suitably impressed. Now: December 25 of what year? Herod, a major player in Matthew’s gospel (chapter 2), died, so far as we know, in 4 BC.
(Via John Salmon.)