No, your GAP insurance would not cover your damages as back up insurance.

Gap insurance's purpose is to pay the difference (gap) between what an insurance company pays out for actual cash value (ACV) when a car is totaled out and the balance of your loan or lease. Without first your car being a total loss no one is paying out ACV for the gap insurance to come into play to pay the remaining amount of the loan. Since your car is not totaled out gap insurance would not pay.

A second issue is there is no primary insurance on your car to pay out ACV for the vehicle if it was totaled in this accident or another one. Since the other party was not insured either than you could not make a third party claim through the at-fault party's insurance either.

Without either you or the at-fault party being insured and your car not being a total loss your gap insurance would not pay out anything normally. What you likely will need to do is take the at-fault party to court to seek a judgment for the cost of your vehicle's repairs if he or she does not pay willingly out of court for the damages you sustained in the accident.