Board Thread:Questions and Answers/@comment-4964532-20140213035348/@comment-3250527-20140213035641

If I remember correctly, it was the Yokohama General Hospital. Otherwise, that would be Asuna's hospital.....

Edit:


 * "Her brother had been admitted to a major hospital in Saitama."

Aside from that reference, there is no name for Kirito's hospital. Unless the anime referenced it, which I cannot at the moment.

Source: Light Novel