I don't know enough about the Civil War to say if this one battle had changed much. But assuming it would have been enough for the South to win the war, or at least to deal out a cease fire with the North, I imagine the following:
There would have been two American states, the USA and the CSA. The Civil War would have ended with a stalemate, and both sides agreed on a peace treaty.
The USA would have become the powerful economic power we know, because most of the pre-CW-industry was located there. The CSA would not have abolished slavery, but due to a lack of industry would not have become as economically powerful and vibrant as the USA. Also, the CSA would have been smaller than the USA, because newly acceding states would have rather joined the USA than the CSA (because the USA are more prosperous, and there are not many slaves in the new territories).
Also, new immigrants coming from all over the world would have preferred the USA over the CSA, for the same reasons, so the USA's power would have grown, compared to the CSA, due to a population growth.
Of course this all becomes even more hypothetical, when we progress further in time.
But I think within the next decades, USA and CSA would have slowly reduced their mutual hatred and animosities due to the civil war, because they would have been natural trade partners: Industrial goods from the USA in exchange for agricultural goods from the CSA. To the degree this would have happened, Abraham Lincoln would increasingly be considered a failed and bad President, because he started a needless war to subjugate a people that is too different from the US to be part of it, but which can be a good trade partner. Yankees and Southeners would feel some sympathy for each other again, but still maintain the belief its better both have a state on their own -- much like Americans and Canadians today.
The CSA would have followed the example of South Africa: Probably, there would have been an abolition of slavery at some point, but very harsh segregation. Blacks would not have been allowed in public and leading positions.
During WW1 and WW2, the USA might have played the same role as it did in "real" history -- because of its economic strength based on industry becoming a military superpower as well. The CSA would have probably stayed aside in these conflicts, maybe found many Nazi symapathizers within their borders, but not joined the Axis, because they would not want to alienate their good trade partner USA, and also would not have had a chance against the USA, military-wise.
In post WW2-times, maybe in the 1960 or a little later, we would have likely seen an emancipation movement by the blacks in the CSA, which would have resembled the end of Apartheid in South Africa. Maybe MLK would have become an American Nelson Mandela.
Once segregation and Apartheid would have been ended in the CSA in the 1970s or 1980s, maybe an increasing number of people would ask for a reunification with the USA. Especially in the CSA, who have learnt to look up to the USA as "big brother", because the USA would have become the real powerhouse in North America and the world, and much wealthier and prosperous than the CSA.
But of course, this is all speculation.