Let's start at the conceptual level. If we think of Hospitals, Clinics, Schools, and Universities as classes of subject matter entities, is there a superclass that generalizes all of them? There probably is. I'm not going to try to tell you what it is, because I don't understand your subject matter as well as you do. But I'm going to proceed as if we can call all of them "Institutions", and treat each of the four as subclasses of Institutions.
As other responders have noted, class/subclass extension and inheritance are not built into most relational database systems. But there is plenty of assistance, if you know the right buzzwords. What follows is intended to teach you the buzzwords, in database lingo. Here is a summary of the buzzwords coming: "ER Generalization", "ER Specialization", "Single Table Inheritance", "Class Table Inheritance", "Shared Primary Key".
Staying at the conceptual level, ER modeling is a good way of understanding the data at a conceptual level. In ER modeling, there is a concept, "ER Generalization", and a counterpart concept "ER Specialization" that parallel the thought process I just presented above as "superclass/subclass". ER Specialization tells you how to diagram subclasses, but it doesn't tell you how to implement them.
Next we move down from the conceptual level to the logical level. We express the data in terms of relations or, if you will, SQL tables. There are a couple of techniques for implementing subclasses. One is called "Single Table Inheritance". The other is called "Class Table Inheritance". In connection with Class table inheritance, there is another technique that goes by the name "Shared primary Key".
Going forward in your case with class table inheritance, we first design a table called "Institutions", with an Id field, a name field, and all of the fields that pertain to institutions, no matter which of the four kinds they are. Things like mailing address fields, for instance. Again, you understand your data better than I do, and you can find fields that are in all four of your existing tables. We populate the id field in the usual way.
Next we design four tables called "Hospitals", "Clinics", "Schools", and "Universities". These will contain an id field, plus all of the data fields that pertain only to that kind of institution. For instance, a hospital might have a "bed capacity". Again, you understand your data better than I do, and you can figure these out from the fields in your existing tables that didn't make it into the Institutions table.
This is where "shared primary key" comes in. When a new entry is made into "Institutions", we have to make a new parallel entry into one of four specialized subclass tables. But we don't use some sort of autonumber feature to populate the id field. Instead, we put a copy of the id field from the "Institutions" table into the id field of the subclass table.
This is a little work, but the benefits are well worth the effort. Shared primary key enforces the one-to-one nature of the relationship between subclass entries and superclass entries. It makes joining superclass data and subclass data simple, easy, and fast. It eliminates the need for a special field to tell you which subclass a given institution belongs in.
And, in your case, it provides a handy answer to your original question. The foreign key you were originally asking about is now always a foreign key to the Institutions table. And, because of the magic of shared-primary-key, the foreign key also references the entry in the appropriate subclass table, with no extra work.
You can create four views that combine institution data with each of the four subclass tables, for convenience.
Look up "ER Specialization", "Class Table Inheritance", "Shared Primary Key", and maybe "Single Table Inheritance" on the web, and here in SO. There are tags for most of these concepts or techniques here in SO.
