There were only conflicting readings of the Constitution, which didn't directly address the question.
That is not an uncommon situation in constitutional law. Usually it's the courts that resolve the question.
In this case, matter didn't come up before the courts before the attempted secession and resulting war.
The Supreme Court did hear a case and pass judgement on a closely related matter after the war, ruling that unilateral secession at will was not constitutional.
Unfortunately, because of the sequence of events people say that the question was decided by the war -- in other words, by force of arms. That's only because there was no definitive legal ruling before the war.
Thank God our founders had no such difficulty with the legality and moral obligation for seceding from a tyrannical government. Or we would still be slaves of Great Britain.