Certainly the celebration of the nativity of Christ our saviour is reason for a feast! It saddens me that Christians might think that Christmas is a pagan holiday though. I'd love to share some insight into why that is not the case, and why it is truly ours.
We do not know what day Jesus Christ was born on, but the date of celebration isn't attempting to be on the same day as his birth. The early church chose to celebrate on December 25th to offset a pagan ritual - it is in opposition to a pagan ritual, in celebration of the true light coming into the world - not in joining with the pagans.
The use of the Christmas tree stems from St. Boniface bringing the truth of Christ to pagans who were worshipping Thor in Germany. He stopped a human sacrifice, cut down their ritual tree, and instructed them instead to have a small fir tree as their holy tree. Their homes that gave them shelter were made of fir logs, and it is evergreen, a symbol of the life that we are offered. There is nothing pagan in this; rather the opposite, the pagan ritual tree was cut down and the truth of the living God was shared with these pagans.
"Santa Claus" stems from Saint Niklaus, which is a translation of Saint Nicholas. St. Nicholas was the Bishop of Myra. He is celebrated on December 6th, inside the nativity fast, particularly for his generosity and mercy done to others. He knew of three daughters of a poor man who had no dowry to bring into marriage. This would have made marriage all but impossible for them and due to their poverty it appeared that if they wanted to eat it would have requried their own prostitution. In the night St. Nicholas went to their home and left enough gold in their shoes to provide money and a dowry. It isn't common in the US but in the old countries it's normal to have children leave their shoes out and to put coins, treats or oranges in them. St. Nicholas is also known for his staunch position against Arias, the proponent of Arianism in the early church.
These things are not pagan, though so much of Christian tradition is lost in the west. These traditions are beautiful and I hope this can offer some clarity as to why they are truly ours, and do not belong to the pagans.