no-fault insurance

Do not confuse medical insurance with auto insurance.

In auto insurance, most states are traditional fault/tort based, i.e. insurance will recover damages from the responsible party and/or their insurance. However, in some state, the medical coverage portion in the auto insurance is no-fault insurance, i.e. auto insurer will cover its policy holder's medical coverage, no matter whose fault it was.

On the other hand, the medical insurance are strictly no fault insurance. The medical insurer cannot sue someone for giving you the bug, cut you with a knife, cause you cancer, to recover their coverage on your treatment.

Of course, you are free to sue someone if they hurt you. But that is for your out-of-pocket expenses and pain and suffering. That has nothing to do with your medical insurer (not) recovering money from responsible party.