Why do you think the Republican Party gave up on supporting civil rights for African Americans in 1877

They didn't.

Explanation:

The Democratic Party was against civil right for African Americans well into the 70's. The Republican Party was responsible for the freeing of slaves, the first African American senator was Republican, and it was also the Republicans that passed the Civil Rights Act in 1964, ending public segregation. In fact, the Democrats even conducted a 14 hour and 13 minute filibuster to try and delay the legislation.