In many parts of the south in the United States there is a widespread belief that the Civil War never should have been won by the north and that they should have been able to leave the union if they wanted to. So yes it would be possible for a reactionary Confederate state to exist today.
The issue to take into account is not slavery - to be clear, in no way am I saying slavery should still exist, and using the Confederate states is a bit inflammatory of an example. Still, the basic idea - that politically and culturally different reasons should be separate entities - has much merit.
States can still secede from the Union based upon the Tenth Amendment of the U.S. Constitution. Powers not specifically enumerated to the federal government are left up to the states. Even if a state secedes, it would have to return all federal property to Congress, which is why the original Civil War started. The Confederate States of America could exist today, just without the violence of a five-year war.
Yes, I think that the Confederate State of America could exist today, because they would have fewer rules and regulations. With fewer rules, their economy would be healthier, and they would probably be the world's powerhouse, not the north. Slavery would have ended not long after the civil war, as people came to realize what an atrocity it was.
When the South tried to secede, it started the Civil War, which helped to keep the United States unified. It seems like the United States is more unified than not nowadays, so I don't think this type of situation could exist. That being said, if it did, I would imagine we would have similar cultures and beliefs, somewhat akin to how the United States and Canada are so similar. But what the Confederacy originally stood for would probably not be popular or supported by most people today, so I don't see the Confederate States, as they were during the Civil War era, realistically existing today.