If the south had won, there would have been 2 countries. The United States of America and the Confederate States of America, and I dare say the CSA would be more apt to stand on their own 2 feet and not turned into a welfare state.
I sincerely doubt it. A nation founded on the idea that any part can secede at will, without consideration for its fellow sections, would have balkanized long since. There would have been the U.S., the Republic of West Texas, the Republic of East Texas, the Commonwealth of Greater Charleston, the Cherokee Nation of the West, the Cherokee Nation of the East, Free Kentucky, etc.
Three problems:
1. As LexBaird points out further down, the Confederacy almost certainly would have balkanized. Had the North let the South go without a fight, the North would have balkanized as well.
2. The Confederacy was a kludge. By that I mean that the confederate government couldn't get out of its own way, and the several states were so obsessed with the idea of avoiding any centralization of power that they carried it to extremes. I recall that the governor of one state demanded that Jefferson Davis reorganize the army so no soldier would be commanded by any officer from another state. I suspect that if anyone tried to tighten the Cofederate organization up, some of the states would have seceded over it, and then we have three countries... Plus, the South would have been mired in poverty with no industrial base. All their value was tied up in slaves. Maybe that wouldn't keep them from being a nation, but they wouldn't be a successful one.
3. As to the welfare state, the modern American welfare state was founded on the conversion of the (mostly Northern) urban black family to welfare in the late Sixties and early Seventies. All those folks' ancestors would have been bound in slavery for some time past the Civil War (at least a decade, as there was NO movement for abolition in the South) and then would have had a much harder time migrating North. So, the North might not have built a welfare state so easily because the racist Left wouldn't have had the false "blacks are still poor because they were disadvantaged by slavery" talking point or otherwise had them to pat on the head, or Southern blacks might have provided leverage for Southern liberals (y'all had them, too) to turn the CSA/Balkan Belt into wlefare states.
Trust me, there's no way that a nation sitting on the backs of slaves was going to do well. Lincoln was right in his Second Inaugural when he said the Civil War was a punishment from God for slavery.