It all depends on what the law as documented in that state provides for in relation to marriage. If it stipulates man and woman, then she will be acquitted. She can say that she did it because of her religious faith, but it won't matter unless she has the documentation to back her decision.

This has nothing to do with civil rights. Civil rights may give everyone the opportunity to get married, but it doesn't give a man the right to marry a pet, a woman to have multiple husbands, or two of the same gender to marry. The court made it's own decision to deem it discrimination not to allow two of the same gender to marry. That same court could also say it's discrimination to not allow Mormons to have several wives, a woman to marry her pet collie. Marriage licenses have always been authorized by state governments, not a federal jurisdiction. I wonder what the court would do if suddenly a state decided to cease issuing ANY marriage licenses. There's not a thing they could do about it. How about if a county told you to go to the next country to get your license, because it no longer issues licenses?
Obama has found it convenient not to obey court orders, so why should states?