Lincoln said at Gettysburg and other places that the Civil War would answer the question as to whether a discontent minority could break from the union and form their country. What was the answer to that question?

Before tackling this question, let's look at what Lincoln actually said, in his exact words. Lincoln said:

This issue embraces more than the fate of these United States. It presents to the whole family of man the question whether a constitutional republic, or democracy, a government of the people by the same people, can or can not maintain its territorial integrity against its own domestic foes. It presents the question whether discontented individuals, too few in numbers to control administration according to organic law in any case, can always, upon the pretenses made in this case, or on any other pretenses, or arbitrarily without any pretense, break up their government, and thus practically put an end to free government.

So Lincoln was speaking not only about the viability of "discontented minorities" breaking away from the United States and forming their own country. On the contrary, he suggested that the American Civil War would answer the question, for the whole family of man, of whether discontented individuals can break down the "territorial integrity" of a government. Now, in the case of the United States, at that time, the answer to the question would ultimately be "no." Discontented individuals proved capable of breaking down the integrity of the government for a period of time, but not sustainably.
However, I would venture to say that Lincoln was being rather hyperbolic in suggesting that the American Civil War could actually answer this question for the whole world, for all time, particularly given the vagueness and breadth of the state he describes. Discontented individuals in the United States proved unable to break down the Union and form their own government permanently, but some fifty years later, discontented individuals in Russia certainly proved themselves able to completely break up the government of their country based on their own beliefs ("pretenses") and form their own government. The territorial integrity of that country, then, proceeded to change several times as other states broke away from it over the course of the twentieth century. Similarly--and perhaps in a situation that more closely resembled the one Lincoln was describing--the discontented minority in the southern part of the island of Ireland broke away from not one country, but two. They were able to break away not only from the United Kingdom, but also from the northernmost part of their own island nation.
So, the answer to Lincoln's question, then, is really this: it entirely depends upon the context, the country, and the time. In the USA in the 1860s, a discontented minority was able to break away from their government and disrupt the territorial integrity of the country for a time, after which the situation returned to the way it had been before. This does not mean that the same thing would happen elsewhere in the world, or even elsewhere in time. Discussions have arisen in the state of Texas in recent years about the possibility of seceding again. Could this happen? The answer is the same: only in the full context of the situation can we say.