Notre Dame is not directly under Church governance at all, since it is incorporated separately and has its own Board of Trustees. In that sense, the Pope doesn't control Notre Dame any more than he controls Harvard. He is certainly not in a position to hire and fire University presidents.
He can't even instantly excommunicate somebody. There would have to be a violation of law, a "crime" so to speak. Obtaining an abortion, or being a material accomplice to an abortion, would be such a crime; but unfortunately, "giving honors to a powerful pro-abort s.o.b." is not actually listed as a crime under Canon Law.
ND's president, Fr. Jenkins, is personally under the CSC Father Superior in Rome. The Superior doesn't "run" Notre Dame either, since the CSC's don't own Notre Dame.
Who runs Notre Dame? The Board of Trustees.
However, that doesn't mean nobody can do nothin' ...
The Bishop can rescind their right to call themselves a Catholic University.
Moreover, Fr. Jenkins' Superior, Fr. Hugh Carey, could get Fr. Jenkins transfered outta there, simply by giving him another assignment. That's a possibility.
There's no guarantee that whoever replaced Fr. Jenkins would be any better. But he might have the fear-of-God put into him by realizing that any more bad publicity and he could be forced out, too.
And Fr. Jenkins could resign. So there's (in my estimation) three realistic possibilities: Bishop D'Arcy can revoke their right to call themselves a Catholic University, Fr. Carey can "reassign" (sack) Fr. Jenkins, and/or Fr. Jenkins could resign. Let's help this happen:
Contact Fr. Jenkins: Call him at 574.631.5000, fax him at 574.631.2770, write a personal email president@nd.edu, or mail your letter to 317 Main Building, Notre Dame, IN 46556
Very Rev. Hugh W. Cleary, C.S.C.
Superior General
Congregation of Holy Cross
Via Framura, 85
00168 Rome, Italy
Or e-mail him: contact@holycrosscongregation.org
And let us pray. With God all things are possible.
I concur with Your thought my email talked about the politics of it Archbishop Burke will get this one resolved.... behind the scenes like a Consligliore...God bless ya!