In many cases, yes, but it depends on the state that you live in. Because of a legal designation called the "doctrine of necessaries”, in many cases spouses are responsible for the other spouse’s medical debt, even if the spouse dies. The largest determining factor is how your state treats the “doctrine of necessaries”.
Common law has historically assigned to a husband the duty to provide for the necessary expenses of his wife, and in most cases, visa versa, especially these days where most households are dual-income and the rights of husbands and wives have become gender-neutral. However, as there is also variation from state to state regarding debt collection, talk to an attorney from your state if you are unsure of your liability. If you are legally separated from your spouse, be sure to address this topic with the attorney as well.