No. The Pilgrims, or Separatists who established Plymouth Colony, did not celebrate Christmas because they could not find any literal references in the Bible that Jesus was born on December 25th (or any other specific date, for that matter). The Pilgrims actually felt that Christmas had become a pagan holiday, as it commonly entailed feasting and drinking to excess.
In fact, Christmas was banned in Massachusetts Bay Colony in 1659, and anyone caught celebrating the holiday would be subject to a 5-shilling fine. It wasn’t until 1856 that Christmas was legalized in Massachusetts Bay Colony.
So what did the Pilgrims do on December 25, 1620?
They spent the day building their village common house, which was 20 feet long by 20 feet wide, and housed the majority of the Pilgrims beginning in January 1621. This common house was built on what is now Leyden Street in downtown Plymouth.