The power to determine whether or not a state wishing you secede may do so is not specifically granted to the Federal Government. The Tenth Amendment states that any power not specifically granted to the Federal Government is reserved for the states or the people. Therefore, the power to determine whether a state may secede belongs to the state government.
Therefore there is a legal argument for secession. Where people have a problem is that we tend to judge historical events through the lens of our own perspective. The US simply wasn’t seen as a true nation by the people living here prior to the war. If you asked a farmer in the Shenandoah Valley where he was from, it would never have even crossed his mind to say “America”. His natural response would have been “Virginia”. That’s where the analogy to the EU comes in. I would guess that nobody in Europe calls themselves Europeans. It just isn’t a country. Admittedly the modern EU is more loosely united than the antebellum US was, but the political and legal analogy remains. It’s kind of an ex post facto justification to condemn the Confederacy for being traitorous since the legality of secession was an open question at the time it happened.
Nowhere in the 10th Amendment or in the Constitution itself will you find the word 'specifically'. As Chief Justice Marshall noted in the McCulloch v. Maryland decision, in the Constitution "here is no phrase in the instrument which, like the Articles of Confederation, excludes incidental or implied powers and which requires that everything granted shall be expressly and minutely described. Even the 10th Amendment, which was framed for the purpose of quieting the excessive jealousies which had been excited, omits the word "expressly," and declares only that the powers "not delegated to the United States, nor prohibited to the States, are reserved to the States or to the people," thus leaving the question whether the particular power which may become the subject of contest has been delegated to the one Government, or prohibited to the other, to depend on a fair construction of the whole instrument." The Constitution gives Congress the sole power to admit a state. Once allowed to join, Congressional approval is needed for a state to split, combine with another state, or alter their borders in any way at all. It is not great stretch to conclude that Congressional approval is needed to leave entirely as well. The power to approve secession is a power delegated to the United States and not the states themselves.
The only way that power could be reserved under the tenth amendment is if the states had that power before the adoption of the constitution. They did not have that power because of their ratification of the Articles of Confederation and Perpetual Union. They agreed to form a perpetual Union(or country if you will) and from that point on could not leave without the consent of the other states.