No. I was honestly always taught that we were a republic, bound by a constitution; a Constitutional Republic... though looking back I had a better than average social science/studies teacher in middle school. That we have functions of democracy baked into the Constitution is incidental.
Just because I have it within me to swim doesn't make me a dolphin, nor am I a kangaroo just because I can get around by hopping from place to place. At the end of the day, I am a human and just because I can tick a few boxes as another kind of mammal, doesn't make me that other thing. Same thing with governments... and most things, if I'm being objectively honest.
I hope this resonates... I can't simplify this any more than that.