One of the charges that Catholics have been known to throw at each other is the epithet "cafeteria Catholic" by which they mean "you pick and choose those aspects of Catholicism that suit you."
Well, apparently in New York there are public officials who consider it okay to be a cafeteria legal officials. They feel quite comfortable deciding which laws and regulations they will enforce/abide by and which they will not. So, the clerk whose actions are described in N.Y. town clerk: I won't sign gay wedding license
feels that since it violates her religious principles for two men (or women) to marry each other she is free to "be the law" in her town. On this basis she feels free not to issue them a license for which they legally eligible.
What if she refused to issue license for people who are divorced? There a lots of Americans who belong to a church that does not allow divorce and remarriage after divorce. Would there be question as to how long a clerk who refused to issues licenses to people who had been divorced would keep their job?