This may give the real GOP establishment an opening to get rid of Agnew once and for all. [As one who was born and raised outside Baltimore and experienced Agnew first-hand as governor (OK, I was 14 when he was elected), and as a staunch TR-style Republican, I have questions as to how well Agnew was qualified to be a full-term president.] My sense is that Scott, Goldwater, Rockefeller et. al. would likely have told Spiro that he'll serve out Nixon's unexpired term but there's not a snowball's chance in hell that he'll get support for a term in his own right--never mind how popular he might be with the hard hats / silent majority.
I'll venture a guess that Rockefeller would emerge as the '72 nominee, divorce baggage and all. Nobody could question his chops given his experience as the governor of NY. Further, with McGovern as the likely Dems' nominee, Rockefeller should be viewed as the only rational alternative. Wallace may make things interesting for the Dems, and may extract his price for silence/not running as a third candidate again (say, naming an old school southern Democrat as McGovern's running mate), which could help tip things further still in Rocky's favor. I don't think Rockefeller would get the 49 state near-sweep that Nixon got, but it wouldn't be very close for McGovern, either.
While I'm at it, maybe Gerald Ford would be Rockefeller's VP: sort of the average guy who can connect reasonably well with the midwest and so on. At the same time, Haldeman, Ehrlichman, and friends would be looking for new jobs, since their boss/mentor is gone and they owe nothing to Agnew or Rockefeller--and vice versa. So: Watergate remains an address and a type of layer cake, and nothing more.
Could be a somewhat-faster winding down of Viet Nam with Rockefeller and Kissinger, but probably not significantly more so than in OTL. Further, Rocky is sufficiently moderate that he'd steal a march on Jimmy Carter or anyone of that ilk. With no Watergate and an end to Viet Nam, chances are Rockefeller gets a second term in '76. That might butterfly away Ronald Reagan, although seismic shifts in a party's center of gravity have been known to happen (see: 1960-->1964 for the Republicans).
This also pretty much squelches any notion of Ted Kennedy running for president (no loss there).
If the GOP nominates Reagan in '80 and he embraces / adopts / tips his hat to the Rockefeller legacy, I don't see him as losing. It might be close if he runs against a national hero like John Glenn, but that's about it: close but not quite. Thus, the assassination of Richard Nixon could keep the Democrats on the outside looking in for some time.