Sorry your insurance is crummy. Mine explicitly says it will pay for my care and that my insurance company can seek damages from those who may be liable and their insurers.
In California, health insurance is not required to and can ask for it back.
From the insurance dept website http://www.insurance.ca.gov/01-consumers/105-type/95-guides/... :
"If I am injured in an accident, does my health insurance pay anything?
Usually, your health insurance pays for your immediate medical care. Then your health insurance company will typically try to get back money from your auto insurance or the other person's auto insurance. When one insurance company tries to get money back from another insurance company, it is called subrogation."
That situation only exists because the other party had auto insurance. If they didn't have auto insurance and you didn't have auto insurance that covered it, your health insurance policy would have covered it.
No, that's not how it typically works. If the injury was caused by car accident, your insurer will likely deny the claim. This happened to me when someone rear-ended me and I went to the ER to get checked out; I didn't want to file a claim with the other party's auto insurance but had to for payment of the ER visit.