May a public university stop a traveling envangelist from giving speeches on the grassy lawn outside the library?

Yes, at least in the 7th Circuit. In a clearly written opinion, Judge Posner dispenses with public forum analysis, saying that this case "falls into a crack between the rules."  A public university can decide who can make speeches on its lawn, if it uses content neutral criteria.  A rule limiting use to the university's own members plus those they invite to speak is sufficiently content neutral.

What is true is that a university that decided to permit its open spaces to be used by some outsiders could not exclude others just because it disapproved of their message. E.g., Rosenberger v. Rector & Visitors of University of Virginia, 515 U.S. 819, 828-30, 132 L. Ed. 2d 700 (1995). But it could use neutral criteria for access, such as that an outsider must be invited to speak on campus by a faculty member or a student group. American Civil Liberties Union v. Mote, supra, 423 F.3d at 444. The difference between invited and uninvited visitors is fundamental to a system of property rights.

Gilles v Blanchard, 200 U.S. App. LEXIS 3234 (Feb. 14, 2007) - has satellite photo of the grounds