Saint Hippolytus in ca. 204 A.D. claimed Jesus Christ “was born in Bethlehem, eight days before the kalends of January [December 25th]” (Commentary on Daniel 4:23). This was before the Romans established December 25 as the feast of Sol Invictus, frequently identified as Mithras, in ca. 270 -275 A.D. If anything the pagans took a Christian holiday and paganized it, not vice versa. Granted, the Church did not universally celebrate Christmas on December 25 until the 4th century.
It should be noted that since the first days of Christianity, the Church has celebrated the Annunciation on March 25, the day the Church Fathers claimed was the anniversary of Jesus Christ’s conception and simultaneously when the angel spoke God’s message to Mary, whom conceived of the Holy Spirit. Since the average pregnancy is nine months long, December 25 — exactly nine months later — was picked for His Nativity.
It is possible that Jesus Christ was born in December or January telling from the counting in the calendar. Saint Luke’s Gospel (written by a Gentile) mentioned that the angel Gabriel came to Mary in the sixth month. The sixth month in the civil Hebrew calendar is Nisan which coincides with March and April. Nine months later would be Kislev which coincides with December and January. This seems to be strengthened by the fact that shepherds were out in the fields with their sheep and the Talmud says the sheep could only be out in the fields during winter and summer.
To be fair, Christians did adopt many winter customs of the Gentiles and the Church was alright with it. Romans celebrated Saturnalia from December 17-23, a winter harvest which consisted in decorating homes with greenery, lighting candles, having parties, taking off from work, giving bonuses, letting slaves take vacation, not to mentioning exchanging gifts.