Can my ex–husband sue if I stop making payments on our home?

Are you obligated to make the payments by a court order, such as an order in your divorce, or some agreement, such as a divorce settlement? If so, then yes, you could be sued, such as for breach of contract violating the settlement, which is a contract. He could sue you for whatever payment amount is ordered or agreed to.
If there is no order or decree specifically obligating your to make payments, but the two of you your ex-husband and you are both borrowers on the loan, then he could in theory sue you to recover your share or portion of the payment e.g. 50%, though this is a more-difficult case for him to make out than the above situation that is because in this event, there is no order or settlement directly ordering you to make the payments, but by taking out the loan with your husband, you jointly with him took on the obligation to pay and if your failure to pay damages him by forcing him to pay more or by causing him to lose the property, then, as a party to the same contract the loan agreement as you, he has recourse against other parties to the contract i.e. you who breach it and injure him. But it's more of an indirect case than in the former situation, since you are not in direct violation of an order or agreement to pay.
So the short answer is yes he most likely can sue you, but the way he'd sue you and the way the case would unfold depends on the circumstances.
Note that if the home was titled in his name only and the loan was entirely in his name and there was no court order or settlement requiring you to pay, but you had simply voluntarily been paying a share, in that case, he would not seem to have any grounds to sue you, because you are not party to any obligations to pay.