IF she had been elected AFTER the SCOTUS ruling then I would say she would either need to resign or be impeached/recalled. But since she was elected and THEN the SCOTUS ruled for Legalized Gay Marriage then I can't see how the law can force her to issue the license being she is refusing to do so on religious grounds. Unless of course the First Amendment has been rewritten lately by way of judicial edict.
I would go a bit further. The constitution explicitly forbids a religious test for public office. Therefore, the requirement to accept the personal authorization of sodomite unions under ones name is unconstitutional.