I am so sorry to hear that this is coming down to this sort of ultimatum.
As a matter of law, she certainly has the right to leave the marital home, whether married or not. Unless under arrest or convicted, people can't be forced to live in any particular place.
But, as for the mortgage, it depends if it is jointly or individually held. If the home was owned by one of the spouses before the marriage and the other one moved in, then unless the home was retitled to be owned by both, it remains the sole property of the person who owned it before marriage. If purchased during marriage, it is a community(marital) property -- and both are responsible for payments on the mortgage.
The thing is, unless there's a court order requiring a spouse who does not want to pay to continue to pay the mortgage on a jointly owned property, then, in the interim, the other spouse has to pay it all to prevent foreclosure
However, that money paid out will be adjusted when marital assets are divided up when the divorce becomes final. There are formulas based on each spouse's contribution to the marital asset pool (salary etc). But, even if the departing spouse refuses to contribute, his or her lack of contribution can be made up when the court adjusts the final division of marital assets at final divorce.
Thus, it is important to keep good records -- if the home is marital property
(not owned by one of the spouses before marriage, or retitled to be jointly owned).
Please let me know if this requires and clarification.
I wish you speedy resolution in this matter!