Discrimination is not perfectly legal in the private sector, neither on a constitutional nor a statutory basis. The Civil Rights Act 1964 states otherwise:
"Title II of the Civil Rights Act of 1964 states the ban on racial discrimination affirmatively, as a right of “all persons . . . to the full and equal enjoyment of the goods, services, facilities, privileges, advantages, and accommodations of any place of public accommodation . . . without discrimination or segregation on the ground of race, color, or national origin.” It defined “any place of public accommodation” broadly, to include all manner of inns and hotels, restaurants, lunch counters, soda fountains, gas stations, theaters, concert halls, and sports arenas that affected commerce, or that were supported by state action. (Title II exempted private clubs that were not, in fact, open to the public.) Title II forbade such discrimination. And it created a private right of action, and authorized the Department of Justice to bring suit, to enforce its provisions."
You might be confusing the private sector with a private members club or a religious institution; the only 'sectors' exempt* from the Act as far as I'm aware. Citation?
*edited typo