This idea had originally been posited on a game I used to play entitled US Government Simulator, or USG for short. I developed, in short, what I thought would happen and initially posted it on my primary blog. Here's a repost of that same thought -- a bit tweaked --
1964 - In a wave of popularity surging from his near death experience, Kennedy wins in a strong mandate for President against Barry Goldwater (similar to LBJ's victory). Kennedy dumped Johnson for Governor Pat Brown of California, who recently defeated Nixon for Governor. Kennedy continues to stand by the Vietnam war, but in contrast to Goldwater sounds much less militant.
1965 - Vietnam continues to escalate, Kennedy's popularity starts to drop. Vice President Brown meets with South Vietnamese leaders in Hawaii promising full support for their independence.
1966 - Vietnam war continues to be mismanaged as it was in real life.
1967 - Nixon begins speaking tour of America and talking about Civil Rights. MLK speaks out against Vietnam war. Secretary of Defense admits that Vietnam strategy has been ineffective before Senate.
1968 - Vietcong sweep in surprising American and allied South Vietnamese forces. Support for the war reaches all time low. Nixon meets with MLK. MLK assassinated (no correlation to Nixon meeting). Nixon v. Brown v. Wallace -- Nixon wins by larger margin than in reality -- Wallace gained support in unexpected regions.
Nixon's Term - Nixon pushes for the Civil Rights movement and deescalates Vietnam as he pledged on the campaign trail. Civil Rights movement moves forward later than normal due to Kennedy's initial opposition to the affair, despite popular belief. The Watergate scandal happens as expected only swifter then in reality. Carl Albert becomes President until 1976. Albert finishes the push for Civil Rights reform
1976 election - GOP primary between Ronald Reagan and Nelson Rockefeller. Reagan barely wins in a hotly contested primary, setting the tone for the Republican party for the next 20 years. President Albert is primaried by George Wallace for the nomination. After losing a few early primaries, Albert comes out victorious and eventually takes on Southern Governor Jimmy Carter as a running mate. Reagan barely wins in 1976 with Robert Dole as his running mate. Iran hostage crisis shortens with the hardliner Reagan handling the situation. The OPEC scenario leads to drilling in Alaska and other parts of the US and a renewed effort into alternative fuel sources. The oil found domestically lets the OPEC boycot having a lesser effect on domestic affairs.
1980 election - Ted Kennedy runs against Jimmy Carter in the Democratic primary, as it does in reality Kennedy's skelleton's surface leading to Carter winning the primary. Reagan uses sound bytes from Kennedy in campaign ads against Carter. Reagan defeats Jimmy Carter in 1980 in a campaign similar to the RL 1980 election. Soviet talks were huge in the early 1980s.
Dole wins on the wave of Reagan popularity in 1984 and continues the policies of Reagan. He pickes Howard Baker as a running mate. By 1988 talks with the Soviet Union had been going so well that Dole swept into reelection by a smaller, albeit still large margin. The Soviet Union began to slowly fall apart and taxes were forced to be raised and, without the charisma of Reagan, Dole was unable to sell it to the American people leading to the same problems Bush had in the late 1980s. Dole gets us into the Gulf War and it ends as quickly as it begins. He handles it about as well as Bush does.
In 1992 VP Baker runs into a problem as eccentric millionaire H. Ross Perot decides to run for President. Popular Southern Governor William J. Clinton sweeps through the DNC primary leading to an interesting Presidential race in 1992. Clinton still wins.
1996 leads to a fragmented Republican primary torn between Pat Buchanan and Senator Phil Gramm. Buchanan barely squeaks by with a primary victory. Clinton wins in 1996 as he did in reality with a slightly larger margin.
In 2000, Popular Maverick Senator John McCain faces off against popular Southern Governor George W. Bush in a GOP primary. The Democrats rally behind VP Al Gore. McCain sweeps in the primaries and picks up Alan Keyes as a running mate. Despite questions about the selection of Keyes, McCain sweeps into office in 2000 with his plain talk defeating Al Gore. September 11, 2001 happens as it does in real life. US sweeps into Afghanistan handling it a bit more manageably than in reality. Intelligence is brought to President McCain that Iraq may have weapons of mass destruction - he waits it out while finishing up efforts on Afghanistan.
By 2003, Iraq becomes a threat and invasion becomes eminent. End of 2003 we invade Iraq. McCain as a wartime President defeats peace-hawk Howard Dean in 2004 in a larger margin of victory than George W. Bush did in reality. McCain is currently our President.