Is an insured driver who is at fault in a car accident liable for damages on a car whose driver has no insurance?
The fact that the other driver did not have insurance is not an issue in determining liability. Since you were at fault in the accident, the other driver should file a claim with your insurance company.
Assuming you are insured, and were at-fault (such as by driving negligently, or unreasonably carelessly) in causing the accident, you are responsible for paying any claims for damages to the other vehicle, whether or not the other driver is insured. When fault is relatively clear, your own insurance coverage should kick in and help pay the expenses to repair the car of the uninsured driver, at least up to your policy limits (that is, the insurer only has to pay up to the amount of coverage you selected and for which you paid; you would be personally responsible for any costs or damage in excess of that limit—which is a reason to not purchase only the minimum insurance).
Damage to another person’s car (or, for that matter, injury to another person) is paid for out of your liability insurance. That is the purpose of liability insurance: to protect other people from your accidents—and therefore to protect you by having your insurance (and not you, personally) pay the cost of your accidents and any damage or injuries you cause. This is why you purchase liability insurance and why the law requires drivers to have it.
In addition, if you have paid extra for collision or uninsured property damage coverage in your own car insurance policy, you can be compensated for the damages to repair your car. Collision coverage, for example, lets you recover the cost of getting your vehicle fixed by filing a claim with your own insurance company, regardless of who was at fault in the accident. (As an aside, your collision coverage covers only the cost of repairing your car, not any injuries. So, if you were hurt in the accident, your collision coverage will not pay your medical bills—but your health insurance should.)
If fault was not clear (e.g., it’s possible the other driver was at fault, at least in part), so that there is a question of who was at fault for the accident, you can still collect for your own car damage from your own insurer—if you have collision coverage—while responsibility for damage to the other person’s car is being determined.
In any car accident that involves personal injury and property damage, your best step is to first report the accident to your insurer, whether or not the other driver is insured. Again, this is why you pay for insurance: to protect you in the event of an accident. Your insurer not only pays for the damage you cause, but also investigates claims and defends you in court as necessary. They are a valuable resource when you are in an accident.