If one party of a divorce stays behind to live in the marital house and he stops paying the mortgage payments, who is responsible for the debt?

If you are on the mortgage note, then you have a legal joint responsibility to pay according to the terms of the mortgage and note.  Absent a court order by the dicorce court requiring a specific party to pay, you really have little recourse.  The mortgage company is not a party to the divorce and its orders have little signicance in who they will seek payment from.