FYI, it wasn't an "old" Roman celebration, it was the feast of Sol Invictus (the unconquerable sun), which was a late Roman cult--dating from 274 under Aurelian. Christianity predates this cult, and Chrysostom in the 380s was saying that the feast of Dec. 25th was established by the Roman Church from the old Roman census records. It is quite conceivable--even probable--that the Sol Invictus cultists took the date from Christianity and not the other way round.
By the way, the earliest mention of both dates occurs in the same place: the Philocalian calendar or Chronography of 354. The Calendar certainly mentions Christ's birthday on Dec. 25: "VIII kal. Ian. natus Christus in Betleem Iudeae." The reference to Sol Invictus is somewhat more tenuous: "N·INVICTI·CM·XXX". Both of these feasts were evidently well established by 354; there really is no evidence to support the idea that the Christians got this date from pagans.
But I agree with the second part of what you said. Theologically, the subject of the feast is the important thing and not the date.
By the way, here’s the calendar of 354:
http://www.tertullian.org/fathers/index.htm#Chronography_of_354
The pagan “birthday of the unconquered” is listed under Part 6 toward the end:
http://www.tertullian.org/fathers/chronography_of_354_06_calendar.htm
The Christian festival of the Lord’s birth is listed in Part 12, Commemorations of the Martyrs:
http://www.tertullian.org/fathers/chronography_of_354_12_depositions_martyrs.htm
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