The issue would have to be appealed and addressed by the courts. Even then, however, there is no clear answer, as case law concerning non-uniform marriage laws is murky. Divorces are an act of a court, and thus divorces are judgments that must be recognized by all states under the Full Faith and Credit clause. However, because marriages are not the product of judicial proceedings, it can't really be argued that the Full Faith and Credit clause would require that a state recognize a same-sex marriage from another state.
Because the Full Faith and Credit clause hasn't traditionally been held to apply to marriages, non-uniformity between states' marriage laws has been addressed with comity principles and conflict of laws principles. Courts have generally held that if two citizens of State A get married in State B, but don't reside in State B, then it's an "evasive marriage" with an intent to avoid State A's marriage laws, and so they uphold State A's refusal to recognize the marriage.
If there's no intent to evade, however, then it's a little more difficult to refuse to recognize the marriage. The most recent restatement on conflict of laws isn't very clear, but based on the older restatements, the state would need a statute on file which states a clear intent to refuse recognition of such marriages, not just a law prohibiting such marriages from forming in their state, and such refusal of recognition would have to be a legitimate public policy interest of the state.
So, a state like Alabama that has an amendment which specifically refuses to recognize same-sex marriages would not have to divorce a same-sex couple that married in another state. The only remaining question that would have to be addressed is whether prohibiting or refusing to recognize same-sex marriages is a violation of constitutional rights under the Due Process and/or Equal Protection clauses.