I agree with Blalron.  Christmas, with Santa, Ruldolph, Frosty, and all that is a secular holiday, which is also (for some) a religous holiday, with the Baby Jesus, Nativities, 3 wise men, etc.
However, we have gone over the whole bit about Christmas being based upon a pagan holiday several times, and dudes- don’t talk like it is an established fact- as it ain’t.  True, Xmas happens to be NEAR the Solstice- but it isn’t on it (4 days off in fact).  Nor was it on it back then. The same goes for it being based upon Saturnalia, as by the late empire, you can’t find ANY date on the Roman calendar that isn’t within a few days of a holiday- and again-  not quite the same day. Besides, Saturnalia was pretty well dead by the time they started celebrating Christmas.
A better case can be made for it being based upon the birthday of Sol Invictus, that is Mithra.  There, we are only off by a day or so (hard to tell with ancient calendars- it depends upon what century it was, you know).  There are many susp similarities between the faiths.  However- ROMAN Mithraism, which is what everyone is talking about- did not reach the Central Empire until well after Christianity, so their faith could have just as easily borrowed from Christianity.  Or both from each other.
In any case, one of the early church fathers came up with a calculated date that figured to Dec 25th- all based upon info in the NT (around 400AD?).  (Of course, other experts came up with other dates, based upon their interpretations & readings- many went for Jan 6th for quite some time). Whether or not he was right isn’t really material- he certainly seemed sincere in his calculations of what he really thought was the “right” day- and none of those calc were based upon Saturnalia or Sol Invictus (in fact, he might not even have known of either).
Barring finding some memo from an early Pope (“Hey, those Pagans are beating us out with this rad holiday, so let’s go with Brother Bart’s calculations of Dec 25th- so we can steal the day from them”), the only reasonable conclusion is that the coincidence of the dates is just that- a coincidence.
(Not to say that many trappings were not out & out stolen from other cultures, such as Mistletoe, Holly & the Christmas Tree, but these are only outward trappings, not something the Church picked out to “rival” a pagan faith)