There is absolutely no question whatsoever that if you sustained a personal injury as the result of a car accident, or as the result of another injury-causing event, the responsible, negligent, at-fault party is required to pay you for the medical expenses you incurred. Economic damages are part and parcel of every personal injury recovery, and medical expenses incurred in the past and reasonably likely to be incurred in the future are a core part of an economic damage award. Injured individuals are reimbursed for their medical expenses in virtually every personal injury claim by the responsible, negligent at-fault party- or, most commonly, by their liability insurance carrier.
Where I see some people get confused concerns the way in which liability insurance claims work. A third-party liability policy, which is the one that would cover your medical expenses after an accident [i.e. other at-fault party's insurance] is not going to cover your medical bills on a periodic, ongoing, or on-demand basis. In this sense, they are completely unlike health insurance companies or first-party insurance coverages [like PIP] which pay medical expenses on an as submitted, on going, as incurred basis.
Rather, a third party liability insurance carrier will pay you for your medical expenses and other economic damages and non-economic damages one time only- as agreed to by the terms of a settlement or as ordered to buy a court after trial.
This is where the confusion comes in, although why a patient would think they don’t have to pay their doctor is puzzling. The liability carrier does not pay you one amount and then separately pay for your medical expenses. It is a single recovery. The recipient of either the settlement funds or judgments proceeds is then generally responsible for accounting for her medical care expenses from her recovery- unless those bills have been paid for by PIP. If the bill is just that- a bill that hasn't been paid from another source- it is due and owing by the recipient of the medical services. If the bill has been paid for by private health insurance or by a state or federal program such as Medicare or Medicaid, then those entities are going to have a right of recovery against the award for amounts they have paid out, as a matter of federal law, contract, or both. In other words, they are entitled to be reimbursed from the settlement or judgment funds for amounts they paid for the medical expenses of the injured person. Through one method or another, the injured person is responsible.