Well, we can have one blockchain for a section per floor per hospital.
And we're going to have CardioHeidelberg token, GastroPitieSalpetriere and so on
Storing it alone wouldn't be that big of a problem, the current system is doing just this at this point but the problem will be the so-called decentralization, how are you going to keep the nodes up, and most importantly who is going to pay for this. Current databases that are running in hospitals here (although not on that scale, as it would be impossible for a hospital around here to have millions of patients) are optimized for this, there are a few copies of the data in the health ministry storage but even with these tiny things there will be no real decentralization but rather giving custody to another entity.
Why do that? No reason!
Another problem I see in this concept is that as we know and use the blockchain everything is final, there is no chargeback there is no block deletion, what will happen in case of human error while handling the data? It's pretty sure they have a way of removing it, so, at this point, what's the point of an immutable blockchain?
In short, rely on a traditional database that can act like a blockchain as close as possible. And we already have tons.