Yes, inspired by the gay marriage debate, although I am interested in the question for it’s own sake, not re. gay marriage. Suppose a state somehow enacted some sort of law (not necessarily marriage-related) that aroused the utter outrage of the other 49 states. Rather than cooperate with it or allow it’s recognition in their states under the full faith and credit rule, people would engage in civil disobedience, judges and police officers would resign rather than enforce it, and public officials would illegally simply refuse to honor it. Something like what happened before the Civil War in regards to the Fugitive Slave law. What conceiveable law could do this, that would not be struck down as unconstitutional at the federal level?
How about the state of Virginia’s response to the 1954 Supreme Court Brown vs. Board of Education decision. The state General Assembly passed a set of laws that included tuition grants so students could attend private (segregated) schools. Prince Edward County then shut down its entire school system rather than integrate, and white students enrolled at a private school – which was built on land donated by a town.
In fact, the entire civil rights movement was based in large part on civil disobedience by both blacks and whites. While the court cases and head-bashing are well documented, there were a lot of cases where local officials quietly gave in, as in the Nashville sit-ins.
But that has nothing to do with “full faith and credit”, the concept that one state’s laws will be recognized by fellow states.
Here is a discussion of the Full Faith and Credit clause. Warning, .pdf file!