Americans call it as Eye Hospital

My guess is that the author did not want to assume that the reader would know what "ophthalmology" meant, so instead of saying "there are hospitals which specialize ophthalmology," he added the "eye" description so that the meaning of ophthalmology could be inferred. It is not necessary or common to refer to a hospital like that.

No.
There are hospitals which have specialties in Internal Medicine or Cancer. New York Presbyterian Hospital has a unit within itself called:

The Center for Liver Disease and Transplantation (CLDT) of the NewYork-Presbyterian Transplant Institute.

But no Liver Hospital.

We tend to name hosptials for people or places, at least, we do in New York City: Lennox Hill, Sloan-Kettering, St. Vincents.

Joe

Thank you both.
So :

Eye Hospital is okay, while Liver Hospital is not?

Everyone would know what you meant with "eye hospital" since there is enough stuff you can do to your eyes to justify a special facility. Saying liver hospital would be ok gramatically, but most people would find an entire facility dedicated to the liver as strange because it's not that big of a specialty. There is nothing wrong with saying "liver hospital" if you have a hospital that just does liver procedures.

Engineer is correct. I think if I was writing about a 'liver' hospital I would refer to it as Hospital for Internal Medicine.

Joe .. Nation

The Mayo Clinic specialises in salads.

England has a Liver Building, (in Liverpool).

Is it a hospital limited to liver disease?

I can't wait for the answer from McTag.

It's a very nice building which contains offices for various insurance and welfare groups.

It's not a hospital.

Joe(that McTag, he's a scamp.)Nation

Americans call it [as] Eye Hospital

BUT

Americans refer to it as Eye Hospital

Nice.
Thank you.

They treat any diseases as easy as treating salads?

It's still a compliment.

A compliment!

Is that a Chinese joke? I like it.

A complement? A condiment!

By the way, I help teach in a literacy class and we have students from China, North Korea, and Vietnam, among others.

(I'm teaching them to avoid AmE like the plague.) (another joke)

Times have changed. You're both right and wrong.

Traditionally speaking, you are right; but when talking it in the backgroud of influence, of science and economy of the United States, you've probably overreacted on AmE and might be wrong. Just think over this example: Microsoft, a new word in AmE, is now a great name; but if it failed to influence the science and economy of the world, it may turn out to be a name going to oblivion. Luckily, it is still a good word - one of American contributions for English language.

Fair enough.