it is, in fact, just as opposite as “private”. probably more so.
the only difference is what kind of organization is in control – and every state (by virtue of being a state) considers itself to have the discretion to defend its control with deadly violence.
if “public” means anything, it means “controlled by the people involved with it”. it means socially-managed, not state-managed. it means that a person with a gun and a uniform doesn’t get to decide things about it.
for the love of life, please stop saying “public” when you mean “state-controlled”.