"Can State governments regulate gun ownership?
Nothing in the Constitution prohibits them from doing so."
In fact it does
Article VI, Section III
This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding.
Why not just get rid of the State Constitutions then?