If you only have liability insurance will it pay your hospital bill?

No.

Liability insurance is only for injuries and damage suffered by others. So if you are injured and it's the other driver's fault, then their liability coverage will pay your medical bills. But if the other driver doesn't have liability insurance, which is illegal nearly everywhere but still pretty common, then you're in trouble. If you're At Fault, then your own liability insurance does nothing.

You could sue the responsible driver to get the money. Since people who don't have insurance tend not to have a lot of spare money, you will likely end up relying on your own health insurance.

You could also buy Uninsured Motorist insurance for bodily injury as part of your car insurance, which pays for your health costs if you get hit by a driver without insurance. Personal Injury Protection insurance can also be bought as part of your car insurance, which pays the health bills of an insured driver, regardless of who is at fault.