Find out about patient records in the Royal Naval Hospital at Haslar/Gosport. Also was this hospital only for the royal navy or did merchant seaman go there ?
I think you will find it was only for royal naval personnel.
Since I am not sure what period you are talking about, it is difficult to point you in the right direction for the records. Older ones may be at the National Archives, since it was a naval hospital. Others may have been transferred elsewhere. They do not appear to be at Portsmouth City Records Office.