Of course. If, as you say, the other driver is completely at-fault, that driver's insurance would still owe for your damages, whether you carry insurance or not.
Look at it this way: You have insurance, another driver backs into you, he's at-fault. You make a claim with his insurance and never even notify your own carrier. It's basically the same thing (though it's somewhat crazy to be driving around without insurance. If you plan to continue this, I hope you have some major assets you can liquidate when the bills start pouring in).