As has been touched upon, the real problems were the tactics of the day, not the weapons. Had Jackson lived, or had Longstreet been in charge at Gettysburg rather than Lee, then there is a real possibility we would live in a far different world today (I, for one, am glad it worked out the way it did).
The more I have thought about it and studied it, a modernized version of a Greek phalanx would have been extremely effective during the Civil War. Imagine if you will, a line of battle armed with revolvers, spears, and bullet resistant shields. Use the revolvers during the advance and take to the spear when distances get close. It would be difficult to inflict enough casualties upon such a formation to have any real effect, and that simple fact would be psychologically devastating to your enemy.
I'll even take it a step further and say that tactical deployment of skilled archers could have been extremely effective in certain instances during the war. A longbow's maximum range is about 300 yards and it's rate of fire is FAR greater than the muzzleloading rifle. Also, broadheads make horrible wounds and if dipped in a nearby pile of dung they would have been almost 100% fatal. Not to mention that bows, arrows, and shields are cheap and relatively simple to build and both sides had plenty of access to the materials required for their manufacture.
In short, while a blowback automatic may have been out of reach at the time, winning strategies were not... At least that is the opinion of General Towe