I can't imagine them denying the application. It's two adult people, with full consent, signing a piece of paper saying they are joining in law. Clerk of the Court asked a couple of questions, we answered I do, he signed the form, and we were married. Pretty easy. Maybe too easy for some...
Funny...in old movies...folks were rushing off to Maryland, Elkton, Maryland, in fact, to get married and bypass the restrictions, waiting periods, blood tests etc, that other states enforced before one could marry, to get married. That changed to Vegas...and their 24/7 on the spot marriages.
Part of the national debate began when former Maryland Atty General Doug Gansler said Maryland was bound by law to recognize same sex marriages through Full Faith and Credit considerations.
You merely assent to a different authority...then follow that up with legal maneuverings to acquire some of the same privileges.