The notion that Christmas was situated on December 25 by the early church because of the date’s connection to the winter solstice and sun worship has no contemporary evidence for it. The earliest we hear of it is in this 12th-century Syriac manuscript:
The Lord was born in the month of January, on the day on which we celebrate the Epiphany [January 6]; for the ancients observed the Nativity and the Epiphany on the same day, because he was born and baptized on the same day. Also still today the Armenians celebrate the two feasts on the same day. To this must be added the Doctors who speak at the same time of the one and the other feast. The reason for which the Fathers transferred the said solemnity from the sixth of January to the 25th of December is, it is said, the following: it was the custom of the pagans to celebrate on this same day of the 25th of December the birth of the sun. To adorn the solemnity, they had the custom of lighting fires and they even invited Christians to take part in these rites. When, therefore, the Doctors noted that the Christians were won over to this custom, they decided to celebrate the feast of the true birth on this same day; the 6th of January they made to celebrate the Epiphany. They have kept this custom until today with the rite of the lighted fire.
It is almost certainly not a true explanation but you still it attested to on the Web.