Probably had a 'Mexican Stand-off' with RFK holding evidence of Hoover's homosexuality and Hoover having the dope on MM's death.
It's generally believed that Hoover had something nasty on every president from FDR through Nixon. It's a safe bet that he had the goods on JFK, even before the Monroe "suicide".
Being a grandstanding transvestite AND the Director of the FBI just don't go together too well, no matter how much dirt you have on "people", or the great unwashed.