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.
"As the Full Faith and Credit Clause requires Illinois to recognize the validity of records and judicial proceedings of sister states, the conclusion will not vary because the post-divorce recorded events underlying this litigation took place in other states than Illinois." "Petitioner and Henzel were married in Nevada. Thereafter petitioner brought her putative husband before the New York court. Petitioner and Henzel subjected themselves to the jurisdiction of the New York court and its decree annulling their Nevada marriage was entered with jurisdiction, so far as this record shows, of the parties and the subject matter." "The New York annulment determines the marriage relationship that is the marital status of petitioner and Henzel, just as any divorce judgment determines such relationship. If the Nevada court had had jurisdiction by personal service in the state or appearance in the case of Henzel and the first Mrs. Henzel, its decree of divorce would have been unassailable in other states. So as to the New York decree annulling the marriage, New York had such jurisdiction of the parties and its decree is entitled to full faith throughout the Nation, in Nevada as well as in Illinois." Sutton v. Leib, 72 S.Ct. 398 (1952)
Actually, it's not. If the public acts and judgments of the States were not accorded full faith and credit in the other States, we would be the European Union, not a federal Republic.