Does campus recruiting decision apply to institutions not taking tax dollars?

Brian Doherty at Reason Magazine suggests in "Recruiters in My Cornflakes!: Time to give McCullough v. Maryland another look" that the actual wording of the recent Supreme Court decision to uphold the Solomon Amendment that bars federal funding for schools which ban military recruiters may actually say that institutions not taking federal money must also allow recruiters on campus. See bottom of page 8 and top of page 9 of the decision.