It certainly does if he contains it to the designated area and they exclude him not because of his viewpoints but because of his behavior. A university, even a public university, has no obligation whatsoever to provide such a designated area, though. That the overwhelming majority of large state universities choose to do so has more to do with academic tradition than legal obligation. A public university that operates entirely inside of a single building can install keycard access and entirely exclude the public.
In fact, the smaller University of Nevada, Reno campus does exactly that, locking all external doors to buildings so that only students, faculty, staff, or other authorized personnel can enter, citing the need "to maximize the security of the buildings, occupants, and building contents."
Courts also definitely have distinguished between the rights of the general public to access the campuses of public universities, including the outdoor spaces of such a campus, as compared to the rights of the students, faculty, etc. ACLU v. Mote (4th Cir. 2005) (holding that political speaker who was not a student could be restricted from political speech in publicly accessible areas of a public university other than those designated for speech). The case implies that the result might be different if it involved a total ban on all speech rather than a reasonable time, place, or manner restriction.
Just because a public university has parts accessible to the public does not mean you can engage in unlimited public speech in all of them.
