I think it depends on the context. Society can be divided into the private sphere and the public sphere. Now unfortunately this distinction has been blurred in modern times, but I think it is pivotal in answering questions like this. That said, I believe that a marriage licensor within the civil context such as a judge should NOT be exempt from performing their duties to the state via their religion. Simply put, if they do not like it then they never should have taken the job in the first place. But I believe that a marriage licenser within the religious context such as a priest should not have to marry homosexuals if it violates his religion. The reason being that this falls within the private sphere--his duties to his Church and the religion it encompasses.
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Agreed. I'd prefer a Church marry a gay couple but I perfectly understand it is well within their right to refuse because they are a non-government entity.