No. This is a popular myth that is actually cited as a fact in some 19th Century sources. It is based on very old feelings against Christmas from the Puritans onward. However it is not true.
Modern scholarship on early Christiantity, such as the chronologists Paul L. Meier (1989) and Susan K. Roll (1995) identify John Chrysostom as the earliest proponent of December 25 as the date for Christ’s birth.
Chrystostom’s chronology was based on the priesthood service dates for the division of Abijah, the eighth of the priestly divisions, in which Zechariah John the baptist’s father served according to Luke 1:5-8. From that, following earlier chronologists like Africanus who suggested March 25 as the date of Christ’s conception, Chrysostom arrived at December 25 for Christ’s birth.
Rather than being based on the Roman festival of Saturnalia (17 December on the Julian calendar), if anything the gradual adoption of Chrysostom’s date December 25 into a Christian celebration soon came into conflict with Saturnalia, leading to Saturnalia being banned.
Some scholars still argue that some traditions from Saturnalia were co-opted from the defunct Roman festival into the Christian festival, but it is difficult to see any hard evidence for this in histories of this period. The earliest celebrations of Christmas appear to have vigorously and deliberately suppressed any practices reminiscent of Roman religion. Which is what we would expect when paganism was still a live force with which Christianity was in daily conflict.