"If Wikipedia wants to have any claim to credibility at all, Essjay should be fired."
Essjay is not an employee of Wikipedia (or, more precisely, of Wikimedia Foundation), but of Wikia. Wikia is a for-profit company independent of Wikipedia.
Whether Wikia fires this person has absolutely no relationship with Wikipedia.
A more relevant question is whether Essjay should resign from the arbitration committee. (I do not think that "fired" is appropriate here, since this is entirely a volunteer position.)
This is not so obvious. The purpose of the arbitration committee is not to judge the content of articles, but to rule on behavioral abuses (people insulting others, or repeatedly pushing propaganda, etc.).
It is therefore unclear how credentials in theology or their absence would interfere.
One may, though, consider that a person who invents himself credentials may have integrity issues incompatible with such a position of confidence.