...Had they left with the approval of the majority of Congess...
First, the States, via the Constitution, allow the Federal Government to act.
Second, if a State requires consent of the other states to leave, the State has no right at all, as the other states will never deem their actions, which give rise to the first State's desire to leave, as offensive.
Third, there is NO PROVISION THAT REQUIRES A STATE TO GAIN THE CONSENT OF THE CONGRESS TO LEAVE THE UNION, THEREFORE THAT REQUIREMENT DOES NOT EXIST CONSTITUTIONALLY. PERIOD.
Fourth, while you deem the south's actions "arbitrary" the southern legislatures evidently did not - they voted to leave. Hence, you prove my second point to you. "Arbitrary" is evidently in the eyes of the beholder.
The legitimacy of Secession is a great litmus test on whether an idividual believes our rights emanate from our Creator or from the Government.
Take your second statement, if a state requires the consent of other states it has no right at all. You can take that to cover any arbitrary action. Having ended slavery, if Pennsylvania does not have the right to allow a runaway slave to live safely within her borders then where is her soverignity? Why should she submit to the laws laid out by other states and be forced to allow marshals to come in and seize that slave? Why? Because the Constitution says so. If North Dakota does not have the right to protect her farmers from wheat grown in South Dakota by slapping a tariff on it when brought into the state then where is her freedom? The fact is that the individual states agreed to restrictions when they agreed to the Constitution. And one of the restriction is that they cannot take an act which affectes the interests of other states unless the majority agree to it.
If the fact that the provision is not clearly laid out means it is not allowed then let me ask you where the provision is to maintain an Air Force? The Constitution does not specifically allow it, we must disband it. Is that your position? If not, if the provision for an Air Force is implied in the 'provide for the common defense' clause then why are no other implied powers allowed? It is clear by implication that the power to allow the change of status in a state is one reserved to the United States.
Finally, it was an arbitrary act. They acted without consideration of the interests of the other states, and wityout consulting them. Approval of one side of the agreement without consultation or consent of the others do not constitute consent.