Well a marriage license is not actually a document certifying that you are married, it is a document granting you the right to get married in the state that issued it. You can't get a marriage license in Indiana and use it to get married in Illinois.

Once you do get married, though, the marriage itself becomes a public record of the state that would be protected under the Full Faith & Credit clause. But as I mentioned in my previous comment, Obergefell v. Hodges had absolutely nothing to do with marriage licensing and states recognizing each other's licenses.