No.... He really isn't.
The season and many of the trappings and traditions around winter solstice predate Christ by centuries at least. Agricultural based faiths (referred to as pagan, meaning 'common' by early Christianity) chose to celebrate this time because of climactic seasonal changes. Literally it marks a point of being "Halfway out of the dark" (in the norther hemisphere at least) as days start to become longer again in terms of daylight hours.
Pope Julius I, for largely political reasons and the fact that the actual date was lost by 350 A.D., Chose December 25'th to 'observe' and commemorate the Savior's birth.
Now having said that, does it make celebration of Christ birth on the 25th invalid?
Not in my eyes, it's still an important event in the Christian faith and whether Observed on the 25th of December or September 23rd remembrance of that baby laid in a manger is important.
Let's just not be so proud as to claim Santa Claus, "Christmas" Trees, Eggnog, Gift exchange, Yule logs, Caroling, etc etc were Christian inventions. ok?