The United States had far too much to lose in a practical sense to let such a thing happen; this is all aside from the nationalistic imperative that seemed to drive Lincoln, as he truly believed that the United States was a special place and concept, and wasn't about to see it disbanded on his watch.

First, look at the political and economic issues at play. Economically, the U.S. wasn't about to let the region driving its GDP just pull up stakes and start their own country. The economic stability of the entire country in the mid-19th century was predicated upon an industrial north, and an agricultural south. They supported each other in a way. Sure, the breadbasket of the mid-west kind of straddled the fence on that dynamic, but at its most basic, the U.S. economy was a house of cards whose foundation was threatened with secession talk. If the south could secede, what was to stop Nebraska, or Ohio, or any other important piece of the Union who suddenly felt like checking out? You can't run a country with important ties to major powers like England and France without stability on the home-front. Although the northern economy could and did survive for a stretch without the south (so long as the mid-west was on their side), long-term, they needed that production. Also long-term, as just stated, once you start the precedent of states being allowed to leave the Union, what's to stop others from doing the same thing, even further weakening the situation?

On a more cerebral level, and only loosely tied to economics, the fact that a country can't or won't put down a Civil War doesn't bode well for its long-term prospects. You're not going to get foreign investments, and again, Lincoln just wasn't going to let it happen. He really did believe in the Constitution and the dream of the U.S.A. as a socio-political concept, and he was willing to fight for its preservation. Also, what is to stop England or France from making encroachments on a divided U.S., one with half its strength gone (if secession had been allowed, that is)? It just wasn't a good prospect for the north.

[Sources - James McPherson, 'Battle Cry of Freedom'; Bruce Catton, 'Mr. Lincoln's Army'; Jay Winik, 'April, 1865']

[EDIT - Spelling]