It was a mite later than that. Even during WW-II the Army maintained a replacement mount station at I belive Ft. Robinson, in Nebraska. Mostly used as beasts of burden by that time, but there were still some mounted calvary at the beginning of WW-II, IIRC. I know horses were used to carry supplies, guns, and even small cannon, in Italy, in the Pacific, probably in Burma, New Guinea, etc.
Besides didn't you ever watch MASH? Col.(Dr.) Potter was an ex Calvery type, from the pre WW-II era.