Wrong. Before Saturnalia, again in Judea, was the tradition that Jesus died on the day he was conceived....March 25. This was the day celebrated long before celebrating birthdays came into fashion. Once birthdays came into fashion, people counted nine months from March 25 and came up with December 25 as the birthday. It just so happened that solstice celebrations were also the norm in some societies, usually week long celebrations which took in December 25. Celebrations, for different reasons, took place at the same times.
Again, this was proven shortly after some theorist came up with the bright idea that Christianity copied paganism. In a few locales it was proven that pagans changed their main day of celebration from the 22nd or 23rd to the 25th to match up with Christian celebration taking place.