if a patient is insured, than an effort will be made to bill the insurance, irrespective of outcome. when you "register" as a patient, one of the many forms signed (besides an attempted release of liability which is usually not binding) is an authorization to contact insurance companies for payment of services render, and further, that you will pay for services deemed by the insurance company to not be covered.
if a patient is uninsured, believe it or not the hospital will attempt to collect payment on the services rendered (my wife has dozens of bills from when her father died and was still getting them even 2 or 3 years after he died, the last one being $25 for the tv and phone in the room!) since no one else was named as "co-responsible" for payment, the letters were simply "returned to sender -- person deceased" 🙂 for a spouse, it may be different, i'd have to check with one of my billers to find out...
ed