Answer
No. liability insurance will only pay for the other vehicle that is
damaged if you are at fault for an accident.
if you have
"full coverage" insurance it will generally cover the damage that you are at fault for on a rental car, within the terms of the coverage. however, most car rental places will also charge you for the rental fees that they could have been getting for the car while the car was being repaired. for instance, if you cause an accident that your full coverage insurance covers the repairs for on a rental car, and it take 10 days for those repairs to be completed, the rental company can charge you for 10 days rental for the money they lost while the repairs were being done.
always check with your auto insurance agent when renting a car to determine if you need to purchase the additional insurance offered.