To answer that we have to look at the faculty handbook. While the section on tenure doesn't discuss an appeal to the President's decision, a section on contracts does discuss appeal a termination. The language of that section is a little vague though. It refers to a termination of a contract, so it's validity to this case depends on whether you consider the tenure track and tenure one single contract or two separate. It's likely that the school will argue that they are two separate contracts.
It may not matter what the administration argues though.
There were serious inconsistencies with the tenure process. The way the school brought in extraneous evidence into the tenure process is a troubling issue. Fr. Holtschneider accuses Prof. Finkelstein of ad hominum attacks on his colleagues and behavior that is not in accordance with Vincentian Personalism. This is troubling because the tenure process is not the process to determine whether or not a candidate has engaged in such behavior. Even if such behavior was admissible to the tenure process, it would have to be proven in a separate process. For example Thomas Klocek faced a harassment board which asked him to remedy certain issues, which he did not. When his contract was under review, they looked at how he did not meet those standards and decided not to renew his contract. Never before was Finkelstein notified that he was being ad hominum or uncollegial or non-Vincentian. Not even in his yearly tenure reviews.
Also it's important to note that this is a case of the administration seeking to extricate, remove, or at best marginalize, the faculty's voice in regards the tenure process. With Finkelstein, the best support he received came from his colleagues. The Political Science department wanted him as one of their tenured friends, and voted 9-3 to have him inducted into the club. The college of LA&S tenure review board voted 5-0 to approve tenure. Altogether, Finkelstein received 16-7 votes in favor of tenure. Larudee was also overwhelmingly popular among her colleagues. She was supposed to be the chair of the International Studies Department next year. Likely her vote was 17-6.
None of this faculty support mattered to the administration though. When Fr. Holtschneider met with students, he explained to them that the votes did not matter to him, what mattered was the reasoning behind them. Who decides what is and isn't good reasoning though? Reasoning is a subjective thing. I don't find it all to convincing that the universe was created by a magical sky wizard in 7 days. Nor do I find it convincing that Sacco and Vanzetti were murderers, and I don't find the anti-Finkelstein and anti-Larudee arguments convincing either.
What I do find convincing is the idea of democracy, and grassroots power. The decision for tenure should not rest with a dictator like Fr. Holtschneider. It should rest with the faculty and students who make up the tenure review committees and make democratic decisions on the issue.
What next? What should faculty do? Many faculty will try to stay out of the issue and say "I don't know enough about the case," unless it becomes an issue over their power. The AAUP has clear guideline on shared governance which DePaul is violating in it's rush to stamp out academic freedom. Faculty will defend these rights. What we should push is more than simply tenure for two professors, but for the faculty, and students to be the final word on who receives tenure and who does not. That means stripping the president of those powers to decide. This would be a monumental step. To reach that it means the the faculty should go through with the appeal next fall. When the administration refuses to recognize it, the faculty must not budge in their demands. They will have to try different ways to route the administration out of office, and to seize the means to grant tenure themselves in regards to funds, etc. They might want to consider going on strike, even if it's a one day or one week strike. All of this might lead to forming a faculty union or even turning the University into a cooperative where faculty, students and staff own and operate the school in a democratic fashion.