In the 1870s the Supreme Court took a series of decisions which weakened the Fourteenth Amendment, thus contributing to the demise of Reconstruction. Several rulings established a restrictive interpretation of the Amendment which limited the power of the Federal Government over the single States on the enforcement of civil rights.
In the Slaughter-House (1873) decision, the Court rejected the argument that personal civil rights were protected by the federal government. The ruling interpreted "national citizenship" as different from "state citizenship", arguing that the former only entailed narrow rights such as the freedom of movement between states. On the contrary, the states had the final words "domestic and local government" and this included civil rights.
In U.S. v. Cruikshank (1876), the Court overturned the guilty verdict of a group of Louisiana whites who had assaulted a gathering of African-Americans. The ruling stated that the Fourteenth Amendment did not grant the federal government the authority to convict the whites. It argued that the power in that sense rested with the states only. U.S. v. Reese paved the way to the disenfranchisement of black voters. It maintained that the Fifteenth Amendment did not give the right to vote to blacks but only established that some arguments could not be used to deny their vote.
I would like to take another angle on this question, as some decisive action by the Supreme Court - or any action at all - could have made a difference in the immediate postwar period. The Black Codes, for example, were sets of laws passed from county to county that put a large number of restrictions on free blacks. It limited their ability to travel, to own property and required they work for a white person. All of these were clearly unconstitutional, violations of the 13th and 14th amendments the country had just adopted. The Court could have struck down these provisions, and the Congress could have funded enforcement of the rulings.
This was unlikely, given the conservative nature of the Court during Reconstruction, and the apparent unwillingness of northern politicians or civil rights advocates to pursue legal challenges to such laws, but it was possible. The death of Lincoln and the abolition of slavery itself took the steam out of most of the civil rights efforts of the time. Nevertheless, the Court could have taken a more active role in protecting the nation's newest citizens.
Now, would this have unified the country itself? Not necessarily, but it wouldn't have divided it further either, as their inaction most certainly did.
In my opinion, nothing. The issues that faced the US were not things that could be solved by Supreme Court action.
During this time period, the major issue that divided people was the rights of the freed slaves. The South wanted to be free of Northern rule, but the Radical Republicans did not want to allow this because of the way the South had been trying to treat free blacks. Ask yourself what the Supreme Court could possibly have done to solve this issue.
My answer is that it could not have done anything. If the Court had ruled that Southern states had to do more to give blacks rights, the South would have resisted angrily (which is what actually happened as the South resisted Reconstruction). If the Court had said the North should back off, the Radical Republicans would have been angry as well.
No Court decision could have forced one side or the other to change their minds or to back down -- the Court does not have that much power.