Just say: Today is Christmas!
Not quite right. Christmas is a season, not a day. Exactly how long Christmas lasts varies slightly from culture to culture, but in the English-speaking world Christmas begins on December 25th (Christmas Day) and ends on January 6th (Epiphany, sometimes called Twelfth Night). (At least traditionally: attitudes are changing.)
Thus, Christmas is twelve days long; Christmas Day, however, is December 25th.
In answer to the original question:
We use "a" (or "an") only when we are mentioning something for the first time, and
we do not know the exact identity of the thing we are referring to. There is only one Christmas Day every year, and so normally it's enough to say, "Today is Christmas Day": it is the Christmas Day of the year the conversation takes place, and so its identity is obvious.
Similarly, "Today is Monday" -- it is the Monday of the week the conversation takes place, and so its identity is obvious.
However, there may be situations where the identity isn't perfectly clear. For example:
"It was a Christmas Day some time in the 19th century." There were 100 Christmas Days in the 19th century, and it's not clear which one it is.
"It was on a Monday in the summer of 1969." There are about a dozen Mondays in every summer, and it's not clear which of these Mondays is being referred to.