You shouldn't have to worry about paying your medical bills when you are hurt in an accident. First, if you have medical insurance, your insurance company has to cover your bills. Don't worry about whether this is fair to it, because you surely have a duty to see that it is repaid from any settlement made on your claim, if you actually wind up making one.
Second, the insurance company of the driver that injured you is obligated to pay your medical bills up to $10,000 under what is known as PIP (Personal Injury Protection) or BRB (Basic Reparation Benefits) benefits, which are sometimes referred to as "no-fault benefits."
Third, if the other driver is not insured or if his or her insurance company refuses to pay your medical bills (which sometimes happens), your auto insurance company must pay them. They have a right to be paid back and can bring suit on their own to collect, if they need to. You don't have to worry about your insurance company being unfairly stuck with the bills; the law protects the insurance companies too.