I notice no one has mentioned that the North didn't manage to invent one either. Frankly there were technical limitations that had yet to be overcome, I believe, not necessarily the ones you might think.
First of all, weapons were already being mass produced on machinery from companies like Pratt & Whitney. Interchangeable parts, adequate finish, the works. But other things like heat treatment may not have quite been ready for automatic weapons. Also, the tolerances may not have been sufficient either. After all, there weren't any bolt action rifles either, were there?
Lots of things were just over the horizon. The Colt 1873 Single Action Army, for example. The first ones were made of wrought iron, I've been told. There were also teething problems with brass cases for rifles. They .45-70 ammunition you can buy now is of a little better quality. In the case of submachine guns or pistols, rimmed cartridges present some difficulties, although if a Colt Government Model can be made to work with .38 Special full wadcutters and every Russian/Soviet machine gun to work with a rimmed rifle round, that's apparently not such a big problem.
Although there was no lack of experimentation, everything is in short supply in wartime and that one was no exception.
Let's see. The war ended in 1865. In less than ten years there were highly successful cartridge repeaters. In about 25 years there were fully automatic weapons as well as bolt action designs that are still in use today. And in 45 years, the Colt .45 automatic appeared on the scene. Some Civil War veterans (on both sides) would have had the opportunity to experience all of those things.
Shoot low, sheriff. They're riding Shetlands!
Underneath the starry flag, civilize 'em with a Krag,
and return us to our own beloved homes!
Buy War Bonds.