Zinn's answer to this question, which is addressed in Chapters 10 and 11 of A People's History, is a resounding "no." In fact, Zinn argues that the Civil War may have even exacerbated class divisions, because it enabled Northern politicians to pass a number of pro-industry (and, in Zinn's view, anti-labor) measures without the interference of the South, which had traditionally blocked such legislation:
Under the deafening noise of the war, Congress was passing and Lincoln was signing into law a whole series of acts to give business interests what they wanted, and what the agrarian South had blocked before secession.
These measures included the protectionist Morrill Tariff, the Homestead Act (which Zinn interprets as being passed for the benefit of land speculators) and a raft of anti-labor laws passed at the federal and the state level. More generally, the period after the Civil War saw unprecedented technological developments in industry that had negative effects for labor. The period after the Civil War, and especially after the Panic of 1873 (itself in part a product of postwar economic contraction) witnessed the rise of class conflict unlike any seen in the United States before. Zinn also addresses (though not as completely) the rise of exploitative labor systems such as sharecropping in the postwar South. So he is clear that the Civil War did not end class division in the United States.
Source: Howard Zinn, A People's History of the United States, 206-246.