Lots of provisions in the constitution allow Congress to pass legislation to define how particular constitutional powers should be executed. The "full faith and credit" clause allows Congress to define by statue how states recognize or not recognize the official acts of other states. In particular the "Defense of Marriage Act" specifically allows states to not recognize "same sex marriages recorded in other states.
Article 4.Section 1
Full Faith and Credit shall be given in each State to the public Acts, Records, and judicial Proceedings of every other State. And the Congress may by general Laws prescribe the Manner in which such Acts, Records and Proceedings shall be proved, and the Effect thereof.
True, but irrelevent. What you cite is is delegated power. "Congress may by general Laws prescribe the Manner in which such Acts, Records and Proceedings shall be proved, and the Effect thereof."
But nowhere are they delegated the power to define the terms in the Constitution.
They were given the power to define who is, or can be, a naturalized citizen. (ie. "Congress shall have the power ... to establish an uniform Rule of Naturalization"), but not to define who is, or who is not a "natural born citizen". They apparently realized that after passage of the 10th amendment, as replaced the 1790 law which purported to do just that.