This woman is refusing to obey a direct court order. We are a country ruled by law, not religion. It is very appropriate to jail someone who refuses a court order, it's not some minor thing, the courts must have to have the ability to enforce the law. People can't just decide to disobey laws because of their religious beliefs and expect the courts to do nothing, that's not how it works.
Many people have used religion to justify all kind of behavior, not allowing women to vote, discriminating because of race, interracial marriages, etc. Once those things were outlawed, they didn't allow people to do decide if they wanted to obey the law or not because of their religious beliefs, this is no different. It is a big deal when an elected official flatly refuses to obey the law, if they allowed her to do this there would be patchwork standards all over, some counties obeying the law, other not. You simply cannot allow that, it would be chaos.
This woman is not being jailed for religious beliefs, she is free to believe gay marriage is wrong, she just isn't being allowed to use her beliefs to break the law by refusing to obey a court order.