As I understand it, any state could have taken a suit direct to the Supreme Court to determine this. No state chose to do so, largely due to internal politics of forcing waverers to join the secessionists. Instead, they chose to wage war on the federal government.
They didn't want a peaceful secession of a few of the Deep South states, which is all they would have got that way. Such a Confederacy would have been unsustainable. They wanted violent emotions to be inflamed which would lead moderates to spring to the defense of their "brothers."
Worked, too, almost well enough to win the war.
Read my post on Ft. Sumter, from a few days ago...it gives a very interesting and neutral perspective.
The Supreme Court can't rule on a sovereign act of a state, which is what secession by a vote of the people of a state represents. I am reminded in this of the 1823 words of John Taylor, leading constitutional scholar and a participant in the ratification of the Constitution.
In the creation of the federal government, the states exercised the highest act of sovereignty, and they may, if they please, repeat the proof of their sovereignty, by its annihilation. But the union possesses no innate sovereignty, like the states; it was not self-constituted; it is conventional, and of course subordinate to the sovereignties by which it was formed.
The sovereignties which imposed the limitations upon the federal government, far from supposing that they perished by the exercise of a part of their faculties, were vindicated, by reserving powers in which their deputy, the federal government, could not participate; and the usual right of sovereigns to alter or revoke its commissions.
Unsustainable? Why? More States joined the Confederacy when Lincoln called upon them for troops to invade and conquer the Southern Confederacy. They were unwilling to join Lincoln in pissing upon the Declaration of Indepedence's concept of political sovereignty.
Funny how the Chi-coms echo Lincoln on the subject of secession and the rights of people to determine their own government and poltical bonds.