Technically, he's right. The Constitution doesn't say who can own a weapon, and it doesn't grant the Right to keep and Bear Arms.
We have that.
It says the Government is to leave the Right to Keep and Bear Arms alone.
The 2nd Amendment doesn't grant a right to keep and bear arms. It severely limits the authority of Government to mess with that Right of The People, as in "shall not be infringed" (IOW left alone, unencumbered.).
Rights can only be legitimately curtailed after someone has been convicted of a serious crime using Due Process of law, in itself limited by the other Amendments (4th, 5th, 14th), but not before.