What does MLK have to do with business regulation? Slavery may have been ultimately rooted in economics, but by the time of the Civil Rights movement, racism was 100% socially motivated, economics had nothing to do with it. In fact, it would have been firmly in the interest of those businesses to be pro-civil-rights, because it would vastly increase their market+customer base, but they made economically infeasible decisions due to social factors.
Well, this is a long explanation: the 1964 Civil Rights Act prevents private businesses from refusing to sell and serve people of a certain race. Same for refusing to hire them. Many Libertarians and Republicans believe that this is violating the business owner's right to hire who he wants. They believe that the free market would take care of things by giving the business owner bad publicity. They opposed (and some still continue to oppose) this, viewing it as the federal government violating state sovereignty.
This doesn't match up with reality. If a business chose to serve whites and blacks in the 60s Jim Crow South, their establishment would get dynamited by the Klan. African-Americans were pretty much barred from using all vital public facilities, and the governments did not spend tax dollars on the "colored" facilities; restrooms and building rooms reserved for black people were decrepit and in serious disrepair. They were restricted to jobs of backbreaking, menial labor such as cotton picking, not making enough money to survive. Many black people had to live off the land, so to speak, since they couldn't pay to regularly upkeep their homes. And African-Americans still had to pay taxes to the State Government, governments which neither protected them nor cared about their rights.
Racism exerted itself as both social and economic oppression.
I'd recommend reading Walking With the Wind: A Memoir of the Movement.
It is written by John Lewis, a member of the civil rights group SNCC (Student Non-Violence Coordinating Committee). His autobiography does a great job of laying out what it was like to grow up in the Deep South, examined the multiple facets and factions of the Civil Rights Movement, the political climate at the time, and shows us how much progress we have made since those troubled times.