SCOTUS has ruled they can. Some people (I'm not a legal scholar enough to understand) seem to think that that was an overreach that will be rectified as soon as orangeman stacks the Court back away from activist lefties. I don't know. I think the devil is in the details.

But you're an educated man, I'm sure you've read in the Constitution where it says that no one may be searched or seized without either their consent or reasonable suspicion of criminal activity. At this point, I think everyone agrees that you need to be checked in public databases to ensure you're not a felon or otherwise not permitted to own the weapon you're attempting to get. Anything further, or registration thereof, is a search of records that doesn't necessarily meet the provisions of the 4th, and certainly not the 2nd.
Personally, I'm a fan of ensuring that I can check to make sure that no black person can get another weapon. Or anyone with a credit score below 700, because those poor people are probably using it to steal something. Or those making less than 150k/year, same reasons. Or having a registry of every Democratic-registered voter's gun, so that I have ample evidence to confiscate a weapon from obviously mentally-challenged citizens.