It's a state matter is basically what they said. However, since some states have legalized same-sex marriage, the others are bound to recognize them.
You bring forth 'full faith and credit'.
Article IV
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.
Interesting theory.
However, this does not apply when states have previously enacted laws to the contrary.
For example, An attorney licensed in one state, may not be recognized as an attorney in another state.
The point is when state law is already in place, federal law cannot supersede state law.
If federal law is in place, then the Supremacy Clause kicks in. So a state may not enact a law where federal law is already in place.
Another example. Horse racing owners, trainers and grooms must possess a license for each state in which he or she works. Why? Because there is no federal law in place that states otherwise.