Since McDonald and Heller, that's basically left up to the Courts to decide.
There are longstanding prohibitions in place, and those prohibitions are to be presumed as Constitutional until decided otherwise.
The part of Section 1 you quoted (the Due Process Clause) does not authorize Congress to mandate Constitutional carry among the several states.
Yes, those are the views of the U.S. Supreme Court. My views on the meaning of the Constitution frequently differ from theirs.