Actually, some theorize that it may have been just the opposite. Depends on whether Constantine or Julian the Apostate officially instituted the "Natale Solis Invictus," December 25th.
Julian's Calendar of 354 institutes a festival "N(atalis) Invicti" listed for 25 Dec. with 30 chariot races. The difficulty is that both Julian and the Calendar post-date the celebration of Christmas on this date in Rome. What we need to see, is data that clarifies whether this was a pagan reaction to Christmas (i.e. Julian trying to coopt an established Christian celebration) or vice-versa.
Others argue "of course the pagans celebrated it first, because it's the winter solstice and they *must* have celebrated it," but there's no incontrovertible written proof either way.
Pinging you to my post 101 on what I thought I remember learning as a child, many, many, many moons ago.