These laws already exist - it's pretty tough to buy a gun from a store in the US if you're a 'Bad Guy'. Criminal record or a history of poor mental health will stop you getting licensed for the most part, for example.
Stores have to perform background checks by law and many (all?) states have a 'cooling off' period designed to stop people getting mad and running to the store for a gun they can use immediately.
Private sales with no requirement for checks etc are legal in some states though.
The NRA says that the horse has bolted - Bad Guys already have guns, and will still be able to get guns, and they're right. Weather it's gangs using street guns or imports, or a 'school shooter' accessing a family member's legally owned guns, they're out there.
I have anti-gun friends in the US who still own them because if they ever get into a situation with someone wishing to do them harm, that person is likely to be armed.
The other issue is the Constitution. Many Americans believe it's their right - God-given - to arm themselves. They see an attack on any part of the Constitution as opening the door to an attack on all of it. First it'll be assault rifles, then all guns, then freedom of speech, etc. It's not that they love guns, it's the principal.
I travel to the US a few times a year and am a member of a gun club. Love shooting on the range, pistols, shotguns, rifles, all sorts. But if I had my way that's where they'd stay.
America will never, ever get rid though. Too much money too deep in politics, too culturally engrained. Too many people willing to accept dead children as the price of freedom.