Burkley headed the Navy-operated Presidential retreat at
Camp David during the tenure of President
Dwight D. Eisenhower, and served Eisenhower on a 1959 11-nation tour of Asia. He signed the President's death certificate. When asked by the
JFK Library in 1967 whether he agreed with the
Warren Commission regarding the number of bullets which struck President Kennedy he replied, that "I would not care to be quoted on that". Burkley, through his attorney, informed the
House Select Committee on Assassinations that he believed that there was a conspiracy involved in the killing of the President. In the 1990s the
Assassination Records Review Board (ARRB), with Burkley and his lawyer now deceased, sought out the records related to Burkley held by his lawyer's firm. However his daughter declined to sign a waiver that would have allowed the ARRB access. After
Lyndon B. Johnson became president, Burkley was appointed his personal physician, and in 1965 he was promoted to
vice admiral. He retired at the end of the
Johnson administration in January 1969. ==Later life and death==