My understanding is that most
countries have a constitution.
OTOH, "Charlie X" tells us, "On Earth today, it's Thanksgiving" (a uniquely American observance) and Roddenberry's story for "The Omega Glory" make me think that American patriotism wasn't really being downplayed for international appeal. Indeed, there are numerous examples of Americanism throughout the series.
And, the Concordance
while it was a fan production, was produced by Bjo Trimble, who evidently had quite a lot of backstage access. And given the near-religious fervor a lot of fans have for this show, I suspect that her use of the word "Constitution" in this context was not without foundation.