Was it one of natural rights/natural law or positivism?

BTW here is the definition for both:

Natural Law.

Positivism.

So, which was it? I mean, it was natural law. The constitution and older decisions clearly show that. (See e.g. Calder v. Bull). 

So, under the natural law thinking, the essential question to whether the south could secede is whether or not their reason for seceding was justified under natural law (or the law of god if you will). So, why did the South want to secede? It was slavery. That was their stated purpose (see article IV of the Confederate Constitution). So, is slavery allowable under the law of God? It is not. Therefore, their secession was not legally justified and bringing them back by way of violence was legally allowed under the prevailing thought at the time.