Article I. Section. 10. Clause 3: No State shall, without the Consent of Congress ... enter into any Agreement or Compact with another State ... or engage in War
Article III. Section. 3. Clause 1: Treason against the United States, shall consist only in levying War against them, or in adhering to their Enemies, giving them Aid and Comfort.
Article VI. Clause 3: The Senators and Representatives before mentioned, and the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution;
All of this became irrelevant once the respective states SECEEDED. The fight settled it, and forcibly brought the rebellious states back into the Union, but the South DID leave. REadmission into the Union being contingent upon the ratification of the Reconstruction Ammendments proves that the South DID leave. If she was out, then the Constitution couldn't apply anymore than it did for England or any other nation. The fight DID preserve the Union, and centralized governmental control over the country.