Uhm no, not quite.
The celebrating of 'Christmas' predates Christianity. The tree, the giving of presents and the place on the calendar were part of the ancient European tradition of celebrating the shortest day of the year, which in our present calendar is situated in the last half of December. Similarly there exists a celebration for the longest day of the year which is still celebrated in parts of Europe.
This celebration obviously wasn't called Christmas because Christianity in turn predates the English language. It probably had some Germanic name which I don't know by heart. When people began to convert the Europeans to Christianity (which proved rather difficult at first, they had to abandon a whole range of gods for just one) it was convenient to tag on to an existing celebration. Some historians estimate the original Christian event was held some four months later, around what we now call April.
Now Santa Claus is a wholly different thing. Santa Claus is an American invention of the 20th century loosely based on the couple of centuries old Dutch tradition of
Sinterklaas. The similarity in the giving of presents and of appearances are striking. The celebration is held on the 5th or 6th of December.
Sinterklaas in turn is loosely based on
Saint Nicholas, the Bishop of Myra, who lived in the 4th century in what we now call Turkey. Saint Nicholas was (in some part of Europe still is) the patron Saint of children and sailors.
OK, OK, I'll stop.