I wasn't trying to argue the point for this case specifically, I was mostly just trying to poke the bear by pointing out that due process does, in fact, apply in situations outside of criminal proceedings. Despite the case I cited being narrown in scope, to my understanding due process rights are actually extended to students of any publically funded school, including colleges and universities.
If this student wants to make a stand that a private school legally did her wrong she still has several potential arguments:
The student's comments about feeling like she was discriminated against due to her national origin could be a prelude to a Title VI challenge. She could trigger due process protection with that, even at a private school. There are quite a few recent cases of students fighting expulsion under Title IX citing lack of due process, and they have won even against private schools. Since it's a federal law, due process applies if the schools want to keep their federal funding.
There is also precedent for expulsion suits being won against private schools based on breach of contract or because the disciplinary process the student faced wasn't fundamentally fair.
I have no idea whether this girl has a leg to stand on legally, but I don't agree with the sentiment that a private school can just do whatever it wants when it comes to disciplinary action. They don't have to follow the same standards as a court of law, but legally they do have to be fair.