The USA is a constitutional republic in terms of its originating structure (even if it is 'de facto' a democracy now in many ways). However, I don't know why you find it controversial that a nation can determine for itself who shall and shall not be citizens of itself.
The English parliament for centuries had statutorily extended "natural born" status to various classes. Congress, per the Constitution, is authorized per Article 1, Section 8, Clause 4 to determine the rule of naturalization, i.e. the power to confer citizenship to others (and the duties, privileges, and immunities that come with it).
But, in principle, any nation has the power to determine who is and is not a citizen of itself. That's kind of the point, is it not?