A variety of reasons.

First, the Confederacy had a hard enough time manufacturing any weapons, and repeating carbines or rifles would have been a bridge too far. There were few, if any, commercially available revolving firearms for purchase in Europe.

The United States had available, however, the Colt Repeating Rifle. The rifle was purchased and issued to some units, however it was immensely unpopular because it was unsafe. The rifle was loaded with paper cartridges which would, under field conditions, allow powder to migrate into recesses in and around the cylinder. The cylinder of the revolver does not actually fully seal, and so hot gasses would ignite that powder, causing chain detonation of all chambers.

This is a risk inherent in all early revolvers, but the difference in how a rifle and a pistol are held means that the consequences of a chain detonation are quite different. The left hand of a rifle shooter is forward of the cylinder, and face of a rifleman is much, much closer than face of a pistol shooter. If a pistol chain detonated, your gun is ruined and your shooting hand may have powder burns. If your rifle chain detonated, your left hand was destroyed and you might be blinded. Of particular concern to the Army, which had to think of sustained battles, was the fact that as residue accumulated, the risk of chain fire would become greater.

Even had those problems been solved, a revolving rifle was still unattractive to the Army because of poor sustained rate of fire. A revolving eifle could fire six shots rapidly, but then all the chambers had to be carefully loaded and capped with percussion caps. This took longer than it would have taken to load and fire six times with a regular issue rifled musket.