Does the Church actually come out and publicly reject formerly Catholic institutions that have lost their way and are no longer fit to be called Catholic? Not usually. Take, for example, Georgetown University. It still publicly proclaims itself as Catholic. The Catholic Church has not publicly proclaimed otherwise.
So how can we, as Catholics, know which institutions are Catholic?
My suggestion would be that we need to judge these institutions based on the teachings of the Church. For example, if you compare the teachings found in the Catechism of the Catholic Church to the practices of Georgetown University, you can clearly see (it would be hard not to see) that Georgetown University is not Catholic in any meaningful sense.