Dr. King used the Constitution to make his case for the civil rights movement all the time and did it in a very intelligent and accurate way. Saying that African-Americans had the exact same constitution rights as every other American including European-Americans and even English-Protestants simply because they were Americans. That the Equal Protection Clause in the U.S. Constitution does exactly that. It protects all Americans regardless of race or ethnicity equally. It doesn’t say that some Americans are more valuable and worthy than others simply because of their race or ethnicity.
With Southern Anglo-Saxon states in America who were govern by Neo-Confederates who decided that they since they lost the Civil War that what they would do now is simply deny African-Americans their constitutional rights and argue they can do that under some bogus ( to be nice ) argument that under the 10th Amendment and what they call states rights that they states can essentially do whatever they want. The problem with that argument is that the Constitution supersede’s states rights. The states have to be inline with the Constitution just as much as the Federal Government has to be.
Another part of the Constitution that Dr. King consistently cited in his argument for the civil rights movement is the First Amendment. The guaranteed right for all Americans to free speech and free assembly in America. The right for all Americans to peacefully assemble together and express their free speech rights and speak out against injustices and anything else that they want to speak out against. Alabama, Mississippi, Georgia, and other states back in the 1960s, obviously had another interpretation of the First Amendment. And simply saw African-Americans as not much more than criminals and terrorists and in some cases still as animals like in the era of slavery and believed they could breakup these protests and deny these Americans their First Amendment rights.
The civil rights movement even if it was considered radical back in the 1960s before racial and ethnic minorities became prevalent in America and before minorities had large numbers and before racism was considered to be evil by in large in the Caucasian community, the civil rights movement was about as mainstream as any political movement we’ve ever seen in America. Because it was about the U.S. Constitution and enforcing it for all Americans. Which under the Constitution itself it’s supposed to be enforced equally for all Americans anyway. And I believe Dr. King always understood that.