If slavery eventually did end in an independent Confederacy, it's likely that it would merely be replaced with a system comparable to or worse then apartheid, slavery in all but name.
I do wonder, though, if the South would ever attempt to start creating bantustans, or if the cheap labor would be too important to them.
I suspect that in the Appalachians - and I guess anywhere that large-scale agriculture or industry wasn't viable - you'd see the establishment of "black-free" zones, while the productive heartlands would keep slaves-in-all-but-name
Not so ninja edit
Why do people always go about with this attitude that in a victorious CSA slavery would end on its own in a few decades?
Wasn't slavery heavily ingrained in the constitution? Even with industrialization I can't see them going "ok, we're now all done with slavery you're all free". I honestly believe it'd have continued in some for or other well into the 20th century at least the mid-point.
I think the feeling is that decreasing economic viability as industrialization occurs (typically this is predicated on the assumption that factory work requires more skill than farm work, which I'm very much not certain is true), combined with foreign pressure would force emancipation eventually.
People tend to look to Brazil, which ended slavery in 1888 after decades of decline. People tend to forget, though, that a lot of this decline happened because of a large drought in the late 1870s that basically destroyed a lot of Brazil's cotton production. Brazil also had emancipationists in the South (both on general principles and due to fear of competition for jobs), which the North did in the US historically, and which the CSA will probably mostly lack. Brazil also always had a much higher population of free blacks, and a smoother climb up for them socially (not smooth by any means, but the racial continuum in Brazil as opposed to the binary definition of race in the US meant that families could claw themselves into whiteness and prosperity over generations.
Still, though, I can imagine France and Great Britain leading the rest of the "civilized world" in threatening embargo on the South unless they get rid of slavery eventually - honestly, probably sooner rather than later. By the mid-19th century, most European bourgeoises and nobles considered slavery to be terribly gauche. Some led loud campaigns against global slavery (leading, among other things, the British declaration that they would no longer tolerate slave trade on the High Seas - which in many ways is only one step removed from outlawing slavery globally, in idea if not in practice). Considering that the CSA's economy is much smaller than the US's, and also is much more dependent on international trade, they'd fold pretty quickly in the face of any real economic pressure...or they'd collapse into poverty. Worst case scenario involves the US being supported by France and the UK at the turn of the century to destroy the peculiar institution once and for all.