No, your liability coverage would not cover damages to your vehicle.

Property damage liability covers you if your car damages someone else's property. Usually it is their car, but it could be a fence, a house or any other property damaged in an accident. It is a good idea to purchase enough of this insurance to cover the amount of damage your car might do to another vehicle or object.

Bodily injury liability covers other people's bodily injuries or death for which you are responsible.

Liability coverage is thus for damages you may cause to others if at fault in an accident, not your own vehicle's damages. To have your own auto insurance cover your vehicle in an accident you would need physical damage coverages of collision and comprehensive. In the at-fault wreck you described you would have needed to have collision on your vehicle in order to make a claim for damages you sustained.

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