Gun owners should be required to have liability insurance.
The legal basis to require gun owners to purchase liability insurance would be the same as that for car owners. You have the right to a gun as long as you cn demonstrate you are responsible for any damage you cause. Having insurance is the way to be a responsible gun owner. Involving large insurance companies in the cause for making guns safer and reducing mass shootings like Columbine, Sandy Hook Elementary, and the Dark Knight shooting would result in a safer society.
Perhaps insurance companies would have much more detailed process to give you insurance, especially if you are buying a semi automatic gun with several magazines. They would charge insurance on every gun.
You would need to show proof of insurance when you buy bullets. The insurance company would have a vested interest to make sure people with psychological issues did not get hold of the gun. Perhaps this would have helped in the Sandy Hook shooting.
Even in the Trayvon Martin case the insurance company would have to pay.
To my understanding if the insured gun causes damages to an innocent person the insurance company would have to pay. If the gun owner was involved in the commission of a crime or felony the insurance company could try to sue the perpetrator to recoup the money. If the person injured was hurt because they were committing a felony then that would negate any requirement for a payout.
In the Trayvon Martin case I would expect they would have to pay out. He would have to have been selling drugs or robbing a house, carjacking, etc. for it to negate a payout.
I think insurance companies have made a big impact on car safety, I see no reason we wouldn't see similar results if they insured guns.
This avoids the entire 2nd amendment argument. You cede the right to own a gun but argue that the gun owners must be responsible for their actions, just like car owners must be responsible for their actions.