No. The general rule is that negligence is required to be held responsible. If I let my next door neighbor borrow my car to go to the grocery store, and he hits someone, I'm not responsible. Unless, the person can prove "negligent entrustment", i.e. it was irresponsible just to let this person borrow my car, e.g. they're a habitual drunk, or blind, or 11.

However, most auto liability insurance covers whoever you permit to drive the vehicle, so the owners policy does typically cover the fender bender on the way to the grocery store.