Hemorrhage is Greek with two of its clearest markers side by side. The rrh transliterates the Greek letter rho with a rough breathing, the same sequence visible in catarrh and diarrhea, and the initial hem- is haima, "blood." The second element is rhagē, "a breaking, a bursting" — so the word means, precisely, a violent bursting of blood.
The spelling has been simplified in stages. English had it about 1400 as emorosogie, which is what happens when a Greek medical term passes through Latin, French and a scribe with no Greek. The modern form was restored by the 1600s to look properly classical, and American spelling has since dropped the ae of haemorrhage back to hemorrhage. A 1906 nursing journal recorded a London hospital man drawing the line at doctors using hemorrhaging as a verb — "odiously bad taste."
hemorrhage (n.) c. 1400, emorosogie (modern form by 17c.), from Latin haemorrhagia , from Greek haimorrhagia , from haimorrhages "bleeding violently," from haima "blood" (see -emia ) + rhagē "a breaking, gap, cleft," from rhēgnynai "to break, burst," from PIE *uhreg- "break." Related: Hemorrhagic . also from c. 1400 hemorrhage (v.) by 1882, from hemorrhage (n.). Related: Hemorrhaged ; hemorrhaging . Slang in Reports: B.I.D. for "Brought in Dead" and "Dotty" are, [Mr. Sidney Holland of London Hospital] considers, permissible expressions, but he draws the line at "fitting" and "hæmorrhaging." Only such terms, he says, should be used as outside doctors will understand. We would say that on a point of such odiously bad taste he might have been much more severe. [Lavinia L. Dock, "The American Journal of Nursing," May 1906] also from 1882