If you call your winter-based celebration a "Christmas" party (or parade, or whatever) rather than saddling it with the "Holiday" moniker, that is absolutely your right as an American.
Of course, should you decide to want to make a nod to inclusiveness and use the "Holiday" term, that is allowed as well.
And if you're not a Christian (or at least someone who merely celebrates the reasonably secular occasion of gift-giving occurring on December 25th) and you feel left out by someone choosing to brand their event with "Christmas" you are free to not shop there or support the owners or vote for those people at the next election. You can write a letter to note your disappointment over their decision. But let's keep the ACLU out of it, okay?
And to every city or shopping center that waited until after Thanksgiving to put up the "holiday" decorations, allow me to applaud that restraint.