It really depends on what you mean by "began as a pagan holiday".

If you mean "were some of the traditions taken from pagan celebrations" then yes, absolutely. The Christmas tree, holly, gift giving and even carols all likely originate from pagan celebrations occurring around the same time of year.

If you mean "was the date of Christmas originally because of the celebration of a pagan holiday" then that is a little more difficult to answer.

There are two main schools of thought on the matter. The first is that Christmas is on December 25th because of a co-opting of pagan religious festivals that fell on this date in the Roman calendar. The second is that the date of Christmas was calculated separately based on various chronological and theological observations.

The central piece of evidence in the first case is the Chronograph of 354 which includes a Roman calendar that documents some Roman festivals from the mid-fourth century. As C. P. E. Nothaft puts it, "the entry for December 25 reads “N[atalis] Invicti C[ircenses] M[issus] XXX” (“Birthday of Invictus, 30 chariot races”). This has been generally interpreted as a feast in honor of Sol, despite the fact that the epithet “invictus” is by no means exclusive to the sun god." The Chronograph also happens to be the earliest evidence of another festival coinciding with Christmas and this has led to their chronological conflation. Because of some of the problems with the Chronograph of 354, including the relatively late surviving copies, the fact the festival calls for chariot races and not sacrifices, and the lack of astronomically significant dates among other Sol Invictus celebrations, "the question of which of these festivals preceded or influenced the other cannot be answered on its basis"

Steven Hijmans makes a very persuasive case that "while they were aware that pagans celebrated a festival in honor of Sol Invictus on that day, this did not concern them, and it certainly did not play a role in their choice of the date for Christmas". Instead he argues that the sun has general cosmic symbolism that the early Christians used as the basis for their own selection of the date and that their nativity celebration is separate from pagan celebrations that took place at the same time.

This leads to the second case, which is that Christmas was calculated to be on December 25th for chronological and theological reasons. The most popular view is that early Christians derived the date from the date of the Passion of Christ which had been established earlier as March 25th. The idea is that the time between Christ's conception and death was exact to the year so, if he was conceived on March 25th then he was born on December 25th. Unfortunately, there isn't much in the way of textual evidence to back up this theory either.

In the end, the problem is that there is probably no right answer. While it is absolutely clear that some facets of what we call Christmas were appropriated from pagan customs, we cannot be certain of why the particular date was chosen. I think Nothaft puts it very well when he says, "one of the most encouraging trends in recent research on Christmas's history is its critical stance towards sweeping narratives and a readiness to consider explanations that are more multi-faceted and to accept a more diverse range of factors than has previously been the case". I think that it was probably a combination of many factors that led to the choice of date and, once it was set, Christmas did an excellent job of keeping itself relevant.

Further Reading:

“Usener's Christmas: A Contribution to the Modern Construct of Late Antique Solar Syncretism,” in Hermann Usener und die Metamorphosen der Philologie, eds. Michel Espagne and Pascale Rabault-Feuerhahn (Wiesbaden: Harrassowitz, 2011) 139–51.

Hijmans, “Sol Invictus, the Winter Solstice, and the Origins of Christmas,” Mouseion ser. 3, 3 (2003): 377–98.

Thomas J. Talley , The Origins of the Liturgical Year, 2nd ed. (Collegeville, Minn.: Pueblo, 1991).

C. P. E. Nothaft "The Origins of the Christmas Date: Some Recent Trends in Historical Research" in Church History Vol. 81, No. 4 (DECEMBER 2012) (pp. 903-911)