I will assume a typical US university context, and as far as I know it does not matter whether it is a strictly private university versus a state-funded university. I assume he is a student ("suspension" only makes sense in that context), and also assume she is a student (as a faculty member, a rather different kettle of fish is implied). There will be some code of conduct that students are required to obey, and the university will take the position that by violating the injunction, the student has violated that code. Suspension is a reasonable response, given that the university would open itself up for a Title 9 lawsuit (for tolerating sexual harassment). The university will almost certainly have the authority to suspend or even expel a student for violating an injunction that involves another of their students.
There may be a requirement for a hearing: you would have to look into the university rules. The chances are low that a university would ignore such a rule (if they did, their action could easily be overturned in the ensuing lawsuit).
If he is going to contest the injunction, that involves the courts, and not the university. The courts will not remove the injunction from "the record". He might get it vacated, dismissed or modified, but there will be "a record". The university is likely to keep records of disciplinary actions against students somewhat private, and again there will be university rules about that topic (some of which are compelled by FERPA). Whether or not he has a right to a university-internal appeal depends on university rules, but you should assume that there is very limited opportunity for appeal. Decisions at the level of the provost / president / board of regents are effectively final, and the only recourse is a lawsuit against the university (where the question will be whether the university has the right to suspend the student, and did they follow their own procedures for suspension).