The National Labor Relations Act creates a right to collective bargaining for private sector employees. It's not a fundamental constitutional right, but it is a right.
I've seen arguments about it being impossible for the state to restrict union activities under the 1st amendment because of freedom of association, and trying to shoehorn it into the 14th under a due process theory (can't deprive liberty without process). These are both restrictions on state activity, i.e. saying that the state can't outlaw unions, rather than saying that a private sector company couldn't just immediately fire anyone who joins a union.
I think you mean "private sector" instead of "public sector" unless you are using a non-standard definition of what public means. In this context public vs private is in relation to the state (government) not the stocks and shareholders.
I've seen arguments about it being impossible for the state to restrict union activities under the 1st amendment because of freedom of association, and trying to shoehorn it into the 14th under a due process theory (can't deprive liberty without process). These are both restrictions on state activity, i.e. saying that the state can't outlaw unions, rather than saying that a private sector company couldn't just immediately fire anyone who joins a union.