The Second Amendment to our Constitution does give individual Americans the right to own a firearm, but the Supreme Court has repeatedly ruled that this is not an unlimited right. Government has the constitutional ability to restrict dangerous individuals from owning a firearm, and can restrict the type of firearms an individual can own. For example, those with a criminal record and those deemed dangerously mentally-ill can (and should) be denied the right to purchase or own a gun -- and individuals cannot purchase or own machine guns, rocket-propelled grenades, or similar weapons.
Currently, we have a country where anyone can buy a weapon (including criminals and the dangerously mentally-ill). At the very least, we should plug the holes in our background check laws to prevent this. It just makes sense, and it is supported by an overwhelming majority of Americans.