A few years ago a group of Muslims objected to handling dog food and pork while working as check-out associates at Target. I said do the job, quit, or be terminated. I feel likewise about the Clerk in Kentucky. Gay marriage is the law of the land. You don't have to agree. If handing out marriage licenses to gay couples is morally objectionable, you can quit. There is no other legal choice and refusing to do your job should be grounds for termination. No one is forcing her to break her moral code. She does have an option, not a good option, but an option. It is no different than the Target Store workers.
When the man signs your check, you do what he says or quit. Life works that way.