The Vatican regularly rejects Catholic candidates whose public status is at odds with Church teaching. For example, Alberto Iribarne was nominated for the position Ambassador for Argentina, but refused by the Vatican because of his remarriage after his divorce.

However, Callista Gingrich is not divorced, still less remarried, and her husband's previous marriage was contracted when he was not Catholic and has probably been annuled. Her marital status is not likely to be an obstacle.

As to adultery, that is a matter between her and her confessor. Anybody who repeats gossip about it is placing himself at odds with Church teaching too, which unambiguously condemns unecessarily making known the sins of another person.