That's exactly the point. It was an act of treason. For rather complex political reasons (the specific history thereof I am not comfortable wading into) it's difficult to recognize it as an act of "war" as, generally speaking, "war" implies a conflict between sovereign states and the refusal to recognize the sovereignty of the South is the casus belli of the "war" here in question.

Political scientists define sovereignty in a bunch of different ways but one of the most accepted is the monopoly on the use of force within territorial boundaries. To that end, when independence is declared it is the people of a region saying "our former government no longer has the right to enforce its laws here; we claim the right to do this for ourselves."

Absent those laws, of course, the former government lacks access to all of the tax revenues, trade revenues, strategic resources, etc which were formerly under their control and this makes for some tense moments. Often, sovereignty being what it is, the solution is to demonstrate that the monopoly on the use of force is still very much in play and this is accomplished by marching some armies around and flaunting the ability to use that force.

If this can't be stopped the independence-declaring folks clearly aren't sovereign over their territory. If it can be then they are.

If this all seems pretty anarchic that's because it is. States (not states like "Pennsylvania" but states like "Germany" and "India") are the only real actors in the international system. There are no authorities with power to supercede them and, to that end, he who has the guns makes the rules.

Recommended Reading: Kenneth Waltz, Theory of International Politics