I think if he sits out 1968, he would be a decent contender for 1972. I don't know much about Nixon's personal popularity at the time, but I think Bobby might have a decent shot at ousting him. Or at least, I'm fairly certain he wouldn't have lost as spectacularly as McGovern did at any rate. There's also an interesting question to consider here; would Watergate turn out to be an even bigger deal, and even sooner than IOTL? Given the kind of sleazy tactics Nixon utilized, combined with his disdain for the Kennedys (and the battle scar that was 1960 to remind him), as well as the factor that RFK would (presumably) be a much bigger threat than McGovern, maybe he's even more aggressive here. Leading to, possibly, an earlier exposure of what he was up to. If it becomes clear what had happened before the 72 election, Kennedy should win easily. If not, and Nixon win reelection, but it all still comes out before 1976, there'd certainly be a case for re-nominating Bobby.
I'm also assuming here that RFK wouldn't be assassinated, or at least not by Sirhan Sirhan. Though who's to say that doesn't come later?