Answer
In summary, what the Bible teaches about the civil rights movement is this: it should never have been necessary. Beginning with the kidnapping and chattel slavery of millions, on through the hateful attitudes that prevented neighbors from using the same drinking fountain, the attitudes and actions that led to a culture where the civil rights movement became necessary were all categorically unbiblical. Christianity and civil rights should go hand in hand. Discrimination based on race or skin color has no place in the Christian worldview.
To begin with, the practice of slavery that introduced millions of Africans to the American South was completely unscriptural and un-Christian. Exodus 21:16 says, "He who kidnaps a man, whether he sells him or he is found in his possession, shall surely be put to death." Several thousand years later, Paul equated kidnapping with lawlessness and rebellion against God’s order (1 Timothy 1:8-10). The New Testament admonitions for slaves to be submissive to their masters does not justify the actions of traders, slave owners, or the government and society that procured and treated slaves in ways directly contrary to Scripture.
After the slaves in America were emancipated, ungodly attitudes and actions toward them continued. There is nothing scriptural about racial prejudice (Galatians 3:28), unfair business practices (Proverbs 20:10), forced segregation within the Christian body (Galatians 3:29), or murder (Exodus 20:13). But human sin continued to shape an abusive society for a hundred years after the slaves were freed.
The goal of the civil rights movement was good and biblical—ensure fair rights and equal treatment for all. Any action that worked against this goal, therefore, has to be considered unbiblical. The Bible not only forbids favor for specific people groups, it forbids unfair treatment of anyone (James 2:1-7).
Thanks to the non-violent policies of many of the civil rights leaders, much of the work toward civil rights was biblical. Free speech is granted to all Americans, and reminding a government and a culture of their constitutional and spiritual responsibilities is good and right. The tremendous effort and patience of civil rights activists to work within local and national legal systems is a great example of positively changing a God-given authority from within. The Freedom Riders, activists who rode buses to challenge states’ segregation laws, were also lawful because the previous year the U.S. Supreme Court had ruled in Boynton v. Virginia that racial segregation on public transportation violated the Interstate Commerce Act. Their endurance through physical attacks and prison is a classic example of 1 Peter 2:20 in action.
At the core of "civil rights" is the God-ordained value of each individual. Every person is made in the image of God. When nations recognize civil rights, they recognize the equality of all mankind. The civil rights movement in twentieth-century America can, for the most part, be considered a good example of encouraging a nation to embody more biblical standards.