The Vatican would have to reject this nomination.  The Vatican does not approve ambassadors whose personal lives are incongruent with Catholic teaching about marriage and family. France learned this when the nomination of Laurent Stefanini was rejected.
Defenders of the nomination of Mrs. Gingrich will assert, correctly, that her canonical situation has been regularized. This only underscores the hypocrisy of the Vatican's rejection of Mr. Stefanini and of the 'pro-family' crowd's embrace of Trump, Gingrich, Giuliani -- all of whom cheated in multiple relationships and who have nine wives among them.
The nomination of Callista Gingrich would be Trump flipping the bird to Pope Francis; but unfortunately, I doubt that Francis will intervene to reject this nomination as he apparently did in the nomination of Laurent Stefanini.