The United States did not suppress the enslaver's rebellion on a whim any more than it did so to end slavery, even though that ended up being how it worked out. There are two huge issues that the rebellion posed to the country, one of which is general to all rebellions and one particular to American political thought at the time. They're tightly related, but it's worth making the distinction for clarity.

First the general: governments which permit rebellions against their authority cease to be governments. The revolting states didn't seek peaceable, mutual separation in a manner like happens sometimes. From the start they asserted their unilateral right to go and used force to seize federal property within their bounds. Put in the most generic terms: a breakaway faction seizes military bases, mints, and arsenals from a state. What would one expect to happen? Furthermore, what the rebels could not take with threats or by exploiting untenable situations -not a lot of defenses around your average customs house when the other side has cannons- they were prepared to directly attack or starve out. I can't think of many situations where a nation of any sort would ever take that lightly, particularly when some subnational levels of government are in charge of the act.

The more specific one is the more compelling, though. We wouldn't care much about a situation like the above happening in somewhere like Nazi Germany or Stalin's USSR except maybe to wish them well. Those were authoritarian regimes drenched in blood so anything that caused them problems isn't going to worry most people too much, unless they're Nazis or Stalinists. The thing is the rebels did not have some kind of atrocities or horridly oppressive state to rebel against. By their own admission, as of November 1860 they lived in the freest polity in the world. Their human property would disagree and Native Americans would want a word too on that point, but white Americans overwhelmingly believed it to be the case.

Such a government could, of course, expire. Americans of both sections were keenly aware that their republican experiment was fragile and essentially alone in the world. Its vast spaces and natural resources might make it hard to conquer from without, and it had survived two wars against the superpower of the age, but they could not maintain it against subversion from within. Fears of this go back essentially to the Revolution. Maintenance of the republic required vigilance. America wasn't just a place, but something white men -and only white men- had to do.

Had the white South alleged a coup or a somehow fraudulent election, they might have acted in the tradition of anti-subversive republicanism. They did not. The dispute didn't revolve around Lincoln's election having even some kind of technical defect or upon a present threat or currently ongoing atrocity, with the lone exception that he might undermine slavery. That in itself is more than enough for the overwhelming majority of the white South to unite against him, which they did, but does not cut it very far outside the slave states.

So far as loyal white Americans are concerned, the country is built on a solemn pledge that Americans as a nation agree, among other things, that they decide their leaders by fair election. Lincoln got a plurality of the votes and won the electoral college. He is, both by formal constitutional standards and the popular democracy adhered to at the time and since, the legitimate president starting in March of 1861. Part of the deal with agreeing to elections is agreeing that your side may lose them, and that is literally the entire grievance of the white South both to Yankees and by the Southerners' own admission.

For the nation to let states bail because the guy they didn't vote for won an election is to end republican government in the very short term. Once the precedent is set, then anybody can bolt for any reason. Every election has a loser to match its winner. So does every congressional vote. Every policy change in the executive. Every anything. If the Union can be dismembered over that, then it is no union and the United States ceases in any sense to be a nation or any species of democracy recognizable to Americans of the era. Every minority would have a perpetual veto over every majority and every reason to use it. The entire premise of republican government would be inverted, even if they could endure it. This strips the state of any legitimacy.

And that would happen. It might not happen instantly, but the interest just of the rump loyal slave states would in itself start the process in motion. Any time they thought something impinged on slavery, they could threaten to join the Confederacy or just strike out on their own. Any other state could follow the same example, whether to join up with the CSA, found their own group, or just go it alone. These states would, individually or separately, lack the strength of the United States and have enemies at their borders on top of it. In short order they would come into conflict or fall to predation by the Great Powers of the era, which is the end of the American experiment.

Or as the tall guy said, the United States fought so

that government of the people, by the people, for the people, shall not perish from the earth.

At the risk of offering a succinct answer, there are two parts to this the abstract and the practical. As for the justification for the US fighting to keep the seceding states in the union you have several solid arguments: 1. there was no constitutional right allowing any state to secede; 2. the US constitution itself was the supreme law of the land with specific mechanisms in place to allow for its replacement, yet it was abandoned and replaced by the seceding states; 3. by extension every citizen within the seceding states was still a citizen of the union, and subject to the rights and protections guaranteed by the constitution, and the federal government duty bound to protect those rights even if a local majority felt otherwise; 4. seceding states contained federal land that belonged to the union.

That last point would turn out to be one of the more important ones in terms of actually sparking the war. Lincoln refused to relinquish forts and federal properties in confederate territories, and refused to accept the confederacy as a legitimate government or to negotiate with them on those properties. This crisis eventually boiled over when the president of the confederacy, Davis, ordered the attack on Fort Sumter to force a resolution there.

Once the two parties had engaged each other on the battlefield the die was essentially cast. The union saw the confederacy as an illegitimate government and once the fighting started there was no backing down until the war had run its course.