The Civil Rights Movement was a movement by the people, not a law to be passed by congress. A result of the Civil Rights Movement was the consideration of many bills passed by congress into law. Among them were the Civil Rights Act of 1964, the Voting Rights Act of 1965, and the Fair Housing Act of 1968.
John F. Kennedy introduced the Civil Rights Ammendment, and Johnson passed it. But MLK was a great leader in the Civil Rights movement for blacks, as well as Malcom X. For the Chicano movement, Cesar Chaves was a hero. Betty Freidam and Gloria Steinem were leaders in the Womans Civil Rights movement, and Harvey Milk lead the Gay Rights Movement.
It is important because the civil war freed the slaves, and the civil right movement helped African Americans gain rights. So if the civil war didn't take place there would be no civil rights movement.To answer your quesition, the civil war was important for the civil rights movement because one freed the africian Americans and one helped them gain rights.