Nixon had four appointments to the Supreme Court (Burger as CJ, and Blackmun, Powell and Rehnquist), so President Robert F. Kennedy could definitely have put his own stamp on the Court.
He ran as an antiwar candidate and, if elected, presumably would’ve governed that way. He would’ve drawn down the U.S. involvement in the war much sooner than Nixon, I believe, without expanding it into Laos or Cambodia. No Kent State shootings, but maybe Saigon falls much sooner than 1975. Detente might not have gone as far as it did under Nixon and Ford, given likely GOP criticism of Bobby for “abandoning our loyal allies in South Vietnam” and suspicions of his maybe thus being “soft on Communism.”
I agree RFK would generally continue LBJ’s Great Society programs. He would’ve been much more pro-environment than Nixon, who signed the EPA act into law without any great enthusiasm. Probably race relations would’ve been better; RFK had strong ties to the black community by the end of his life. Anti-Mafia investigations probably would’ve been ramped up; the Teamsters would definitely have had a tough time with a second President Kennedy. He might even have authorized a second investigation into his brother’s assassination; he told aides not long before he died that he wasn’t entirely convinced by the Warren Commission report.
Nelson Rockefeller was still a (relatively) recent divorcee around that time and, even in 1972 or 1976, I doubt he would’ve been able to win the GOP nominee, even without the Attica prison riots. He had a pretty hard time even getting confirmed as VP when Ford picked him in late 1974 IRL, after all. Nixon, having lost in both 1960 and 1968, would be finished politically. Reagan would’ve maybe peaked sooner; Ford would’ve stayed in the House. Maybe Bob Dole or Howard Baker come to the fore sooner. I see no reason why Bobby couldn’t have been reelected in 1972, assuming the economy was doing OK as it was IRL, and assuming the Vietnam War hadn’t come to too disastrous a conclusion by then.
Tom Wicker, VERY early in the 1968 campaign, wrote a paperback what-if novel in which LBJ dumped Humphrey from the ticket, swallowed his pride and picked Bobby as his running mate when he sought reelection that year, already 'way down in the polls. Reagan and Nixon deadlocked the GOP convention, which finally after many ballots turned to NYC Mayor John Lindsay, renowned as a moderate Republican who kept a lid on his city during rioting elsewhere that year and was noncommittal on Vietnam. Sen. John Tower of Texas (yes, the same guy rejected IRL for SecDef in 1989) was foisted on him as a running mate by the GOP poobahs. After a hard-fought campaign, Lindsay just edged LBJ in the popular vote, but they tied in the Electoral College. The House of Representatives tied, too, so the Senate, with a healthy Democratic majority, selected Bobby as VP. When Jan. 20 arrived and the House, exhausted, was still deadlocked, Bobby moved into the White House as Vice President and Acting President, much to LBJ’s chagrin. Crazy novel, but fun. No more unlikely than what actually happened in 1968, you might say…!