It's called the "full faith and credit" clause of the U.S. Constitution. States are required to provide "full faith and credit" to each other's legal proceedings and official documents, unless a particular state has a strong public policy interest in not doing it. Exactly what that means is something that's murky. There has not been a lot of litigation on this question because it's rarely an issue. As far as I know, the last major case kind of related to this was Loving v. Virginia, where Virginia both refused to recognize a District of Columbia marriage license given to an interracial couple, and threatened to imprison the couple for violating the state's ban on interracial marriage unless they agreed to leave the state. (Exile in other words.) But the Supreme Court didn't get into the full faith and credit clause -- they tossed the Virginia ban on interracial marriage as unconstitutional racial discrimination. The most recent Congressional actions on this have had to do with child custody arrangements, and in the Defense of Marriage Act, that allows states that don't recognize same sex marriage to maintain that policy even if they have gay or lesbian residents who were married in another state. This is possibly unconstitutional, but has not been litigated conclusively as of yet.