You write: "I think that there is no way that the institutional Church can determine whether or not an apparition really occurred. "

If any experience of the divine (visions and other forms of private revelation) is aspiring for wider reception, it is assuming a certain prophetic role by addressing the community. 

Prophets in the Jewish-Christian tradition are never self appointed. The prophetic voice is such because it has been recognised as such by the community (that always involves a certain institution). 

Therefore the community, the institutional Church not only can, but has to either recognise it as such or not.