This question has been discussed before:

• What are the historic facts about December 25th and Christmas?
• Why is Christmas celebrated on December 25
• I've often heard that the date many Christians celebrate Christmas was set to coincide with the winter solstice. If true, was there ever a Christian holiday near the summer solstice?
• Happy Festivius/Winter Solstice/Christmas/Yule, etc., etc.! Every year it comes up, so let's clear it up! What are the truths and myths behind Christmas?

The consensus on /r/AskHistorians seems to be that while Christmas does use some Pagan traditions in how it is celebrated, the choice of date was mainly based on the idea that Christian saints die on the same day of the month that they are born, and the dating of Christ's death to March 25th.

In Sweden, Christmas is still called Yule (spelled Jul in Swedish) after the ancient pagan tradition. It's similar in many other languages, as far as I know it's only in the latin languages that christmas is directly related to Christ/Birth.