Did the Civil War in the USA settle the matter of secession once and for all?
Not the Civil War itself, but the court case that followed.
In Texas v. White, there was a dispute in 1869 over who owned bonds issued to the state of Texas after being sold by the state legislature to fund the Confederate states during the Civil War. The majority opinion written by Justice Chase stated:
The Union of the States never was a purely artificial and arbitrary relation. It began among the Colonies, and grew out of common origin, mutual sympathies, kindred principles, similar interests, and geographical relations. It was confirmed and strengthened by the necessities of war, and received definite form and character and sanction from the Articles of Confederation. By these, the Union was solemnly declared to 'be perpetual.' And when these Articles were found to be inadequate to the exigencies of the country, the Constitution was ordained 'to form a more perfect Union.' It is difficult to convey the idea of indissoluble unity more clearly than by these words. What can be indissoluble if a perpetual Union, made more perfect, is not? --Texas v. White
In the Supreme Court's opinion, no state can secede from the United States once joining it. The Articles of Confederation were the basis for the Constitution, and the Civil War confirmed and strengthened the union.
Secession is something that can never really be resolved. In order to be a separate country/nation, you must have sovereignty over the land which requires you to not be beholden to the laws of the country you seceded from. What the Civil War established is that the U.S. is willing to fight a war to prevent secession, and winning that war would be required for any attempted secession. In practical terms, the fact the U.S. considers secession illegal is irrelevant unless a state attempts and fails to secede, because if they succeed they don't care what the U.S. thinks.
A successful secession would also lead to some very complicated international politics issues which would be similar to those Taiwan is and has been facing. Any state or states the would try to secede would have to find allies to officially recognize them as separate from the U.S.