Must the surviving spouse pay remaining medical bills if there are no funds available?

I would let an estate lawyer look more closely at this. Generally, a deceased person's debts become the debts of his or her estate only, and are not the responsibility of his or her heirs. Exceptions can occur where the next of kin sign for treatment as a guarantor of payment for medical services. Another exception is where the deceased person was receiving Medicaid, Medical Assistance, etc. In that case, a medical recovery against the marital estate could happen and there are exceptions and exclusions to this which are too complicated to address in this email. I would not pay a single bill until this is reviewed by an estate attorney.