Which is impossible. How do you eliminate the private sale of...anything? Selling a gun to a buddy of mine is just as easy as selling a computer or a TV to someone.
"Gun show loophole" is a scare-mongering term and makes gun rights advocates suspicious of gun control advocates. If I'm not mistaken, the vast majority of gun sales at gun shows are actually done by licensed gun dealers, where a federal background check is required and will be completed. A private sale, which is what gun control advocates want to require background checks for, can occur anywhere.
I can call a buddy of mine right now and sell him a gun, just like I could sell him a recliner or a TV. How are you supposed to enforce a law that requires a background check to be performed? That's where the fear of such a proposal comes in; how would it be enforced? A national gun registry? I'm not sure; you tell me, but I see what it alarms gun rights folks.