No, expecting not to be discriminated against in hiring or in the legal system based on race is a "civil right". The definition of "privilege" that pops up on a Google search is:
"A special right, advantage, or immunity granted or available only to a particular person or group of people."
Martin Luther King didn't do his work to attack "white privilege"; he did it to ensure that "civil rights" are available to everyone, regardless of race or (as it has grown over the years) sexual orientation, age, or other protected classifications. He wasn't looking to make whites be deprived of some special status and be treated the same way as blacks. He was working to make sure the opposite occurred. Sure, that ideal doesn't exist in practice. Society is composed of people, and people have the inherent ability to hate and discriminate. The goal has to be to improve our society until the civil rights that are spelled out in the Bill of Rights, Supreme Court decisions, and laws, which legally belong to us all, are practiced that way as widely as possible.
You'd have to be blind or obtuse to think that civil rights, as practiced, are applied equally in our culture. You'd have to be equally blind or obtuse to deny that we've improved dramatically in the last few decades. We have a black president, a woman is the Democrat Party's candidate for president, Oregon has an openly bi-sexual governor and there are minorities of all types gaining positions of power and responsibility throughout government, legal, and corporate spheres. We've got a long way to go, but we need to keep pressing for everyone to be treated by the system in accordance with the rights they're due. It's a matter of honor and integrity, not privilege.