This may come as a shock to many of you, but I believe that this woman should be fired.

Businesses (including public/government offices) should make reasonable accommodations that allow you to perform your job functions whether they be for religious or physical reasons. But if your job is to do something you object to on religious grounds, you need to find another job. It's no different than the case of a lesbian woman who doesn't want to teach traditional Catholic values at a Catholic school.