They had NO LEGAL grounds at all. That was the whole point. Not only was there no procedure for leaving the States had no grounds to leave. The Union had bent over backwards accommodating the South since the Union was formed. And in the 1850's had even created new legislation to appease them and they had won Supreme Court Cases in regards slaves and the North.
They had the same legal grounds to secede as the Republic of Texas has to secede from Mexico or that the colonies had to secede from the British Empire. What part of the Constitution did the South violate by leaving. Where was law prohibiting them to do so? There wasn't. So in the absence of law, there was no illegal act.
Texas v. White, though ruling that the States had no right to secede(after the fact with a radical Supreme Court BTW), it allowed some possibility of divisibility through revolution, or through consent of the States. This is the only case that deals with seceesion and its legality and it extremely vague and was meant to be. The only thing that made what the South did illegal is its military loss. Had they won or just negotiated a peacful end, then we would be looking at ths a whole lot different.
But let me lay this on you. If the States had no right to secede to begin with, then why did Congress have to readmit them on litmus? If they had no right to secede, shouldn't they have already been in the Unon to begin with(as
Texas v. White rules) and not eligible for readmittance as they were already members? Fact s they legally seceded, they lost the war and the Union had to readmit them on litmus.
Sit down old man, I do this for a living.