on this issue, Romans 1 is much more on point. Perhaps you should read it sometime.
Would you sign your name on a gay marriage license. Would you put your personal seal of imprimatur on such a document? Would that be God's will?
Romans 13 definitely is not a carte blanche license. Once that is granted, then one understands its role. This is talking about what proper government should be like. It has become improper here and therefore God rather than man should be obeyed.