I think she should resign for her own good, but it's really not a big deal. If I were the Federal judge in this case I would have thrown it out of court and told the plaintiffs to take their case to a venue where it belongs: the Kentucky court system.
Kentucky actually has a criminal statute on its books that covers a situation where a public official refuses to do his or her job. It really sounds to me as if that's the only legal process that might apply here ... and since it appears that there is no applicable Kentucky marriage statute here, there likely couldn't be any prosecution anyway.
"Resign or be persecuted" sounds like a purge to me.
You need to read more, there is an applicable Kentucky law, passed in 1998.