The clause is self-executing, no Congressional legislation is required.
Congress has legislated under the clause, however, in the Defense of Marriage Act, which is designed to block the scenario I outlined. I just don't see how the DOMA can survive the reasoning of the recent case. On the other hand, O'Connor doesn't see the need to employ consistent reasoning from case to case, so DOMA might be safe after all.
Seems like a pretty superficial difference. A marriage license is just as much a public record as a marriage certificate. In both cases they're essentially just statements that the law applies to the persons in question differently than it otherwise would. And it's up to the individual states to decide for themselves just how the law would apply (if at all) in such cases
The clause is self-executing, no Congressional legislation is required.
It may well be a matter of custom that states recognize marriages contracted in other states, just as they, in the vast majority of instances, recognize marriages contracted in foreign countries. But to my knowledge there's never been an instance wherein a state was challenged in federal court for refusing to grant legal sanction to a marriage from another state that failed to comply with its own marriage laws.