Does liability insurance cover insurers car if other driver is uninsured?

No, liability insurance only covers the other vehicle if you are At Fault for an accident.

Coverage for your own car if the other driver is uninsured would come from one of two places:

-your collision coverage, this would be the case regardless of if you or the other driver is at fault

-your uninsured motorist coverage. This would be the case if the other driver is at fault. This is usually a separate part of the policy, and may or may not be included automatically in your policy depending on the state. I would suggest you either check you policy coverage, talk with your insurance agent or talk with your insurance company.