If she had campaigned for change, that would be drawing lots of attention and much more akin to "preaching." Letting her subordinate do the licensing is a much less preachy approach. Just refusing to do the licensing is a much less preachy approach than what you are suggesting.I'm sorry, but any time some (legal) aspect of your job is troubling you and your method of dealing with it is to just stop performing that duty, you're making a statement. If she felt issuing licenses to same sex couples was wrong, she could easily have done so anyways while campaigning to get the law changed. Or she could have quit her job and taken one where her beliefs weren't directly opposed by her duties. But she didn't. Instead, she decided that her personal beliefs could define her job, which sends a message even if that wasn't exactly what she was trying to do.
Also, I am sure when she first took up the job she did not envision the day where she would be marrying same-sex couples; she did not take the job with this particular dilemma in mind. The job duties changed around her.
At any rate, if they choose to fire her, they would be in their right to do so as she is not fulfilling her job duties. However, she is also within her right to stay there until they do so, just like any other job.