OK, it's not a good joke, but the man was busy. It was Feb. 8, 1968, and Martin Luther King Jr. and the Southern Christian Leadership Conference were in the thick of planning the Poor People's Campaign, which would culminate in a tent city going up on the Mall in Washington. It was also just two months before King was assassinated in Memphis.
Though this was well into the Johnny Carson era of Tonight, Carson was off that week—Harry Belafonte, singer, actor, activist, and confidant of King's, served as guest host. Here's an SCLC release announcing King's appearance. Also on that show: Nipsey Russell and Paul Newman.
The video above appears to be the only clip of King's Tonight Show appearance to survive, at least outside of the NBC vault. But if you're into him making the talk show rounds, here's his full appearance on The Merv Griffin Show in 1967.