He can't sue you for the mortgage payments. But there might be issues of reimbursements depending on whether the property is community or separate property. Generally any property bought during the marriage is community property regardless of title designation. Additionally, if he owes you spousal support and both are still staying on the property, it is possible that the payment of the expenses of the house while both are living there could be credited in part, towards spousal support. But there are alot of facts that need analysis in this scenario. You need to see the help of an attorney on this.
Divorce is the process of formally ending a marriage. Divorces may be jointly agreed upon, resolved by negotiation, or decided in court.