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.
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."
Actually your insurer would likely go after his, assuming he had it, to get their costs mitigated.
I suppose that could happen to some, but my insurance pays for all care after I satisfy the flat out-of-pocket requirement.
I suppose my friend would be liable for the out-of-pocket cost, but that would typically be only several hundred dollars--a fair amount of money, but not much relatively speaking if this is a wreck that caused injuries.
That covers most garden-variety car wrecks, so there is little need to be concerned about my friend's insurance. I suppose if I were really on top of things, I would ask my friends for proof of insurance anyway, because maybe I would want to sue him for pain and suffering, which his insurance might pay for (my health insurance would not pay for that, or for lost wages.)
Never seen "riding around in someone else's underinsured car" insurance. Typical umbrella policy wouldn't cover that unless I'm negligent. I suppose I could get disability coverage which might help protect against the lost wages from getting injured in someone else's car.