There are 60 to 65 million gun owners in America.  Licensing all of them would, in the words of the National Rifle Association, "require the creation of a huge bureaucracy at tremendous taxpayer cost, without any tangible anti-crime benefit."

The most common result of licensing and registration in other countries has been the eventual confiscation of all privately owned firearms.  Great Britain provides a perfect example.  Those who feared licensing would lead to confiscation were dismissed as paranoid, but once the public got accustomed to the idea of licensing guns, the government slowly tightened the licensing requirements until they'd effectively banned all firearms.