If the move would have been approved, they probably would have renamed the team the LA Angels. Also, there probably would have been a movement to get a team in SF sooner, like an expansion team or a team that moved there. Maybe Tony Morabito, the first owner of the 49ers, who tried to get an NFL team in SF since 41, gets an expansion baseball team and names them the 49ers.
How this would affect the future:
1. The Giants would have moved to Minnesota instead.
2. O'Malley, with no enticement from LA, and the Giants gone, decides to take the ballpark in Flushing in the late 50's since Moses can't give him the land for the dome in Flatbush.
3. The A's move to Baltimore instead, and the Braves still move to Milwaukee, then Atlanta.
4. The original Senators move to Dallas in 61 instead. The AL expands to include the new Washington Senators and Seattle Rainers, and the NL expands to include the Houston Colt 45's(later Astros) and Los Angeles Stars(to get an NL presence in LA to go along with the AL Angels).
A multi-purpose cookie cutter stadium will be finished in 1970 in Arlington to house the NFL Cowboys and MLB Rangers.
5. Seattle gets a stadium built by 1967, and attracts an NFL or AFL expansion team to go along with the Seattle Rainers baseball team.
6. Baseball expands in 71. In the NL, the Buffalo Bisons and San Diego Padres are added. In the AL, the Kansas City Royals and the Milwaukee Brewers are added to take the place of the Braves.
7. Two more teams are added, and will start play in 79. The Toronto Blue Jays will be added to the AL. In the NL, Denver and Tampa battle it out. MLB chooses the Denver Bears. Marvin Lewis, who failed to bring the White Sox to Denver in 78, will be the owner.
8. No more expansion happens until the early 90's. The cities up for it are Phoenix, Miami, Tampa, Vancouver, and Montreal. Phoenix and Miami are selected to start play in 1995.
There are all sorts of problems with this
1) The Giants move wasn't a done deal, and is very possible after the site of Shea Stadium is turned down by O'Malley, Robert Moses approaches Horace Stoneham for the site instead. Stoneham's flirting with Minneapolis was only semi-serious he was hoping to convince New York to build a new stadium, and if that fell through the move would commence. Suddenly there is a new Stadium planned to be built in Queens that has no tenant, Stoneham won't need to move the Giants after all, and is likely to get a co-tenant in the Football Giants, thus Shea Stadium becomes known as Giants Stadium.
2) O'Malley is still very likely to move the Dodgers, however there are numerous other likely locations, Houston is the most probable, it is large, growing, and has a Stadium that can easily be converted for uses in the Majors, nor is it too far away for other owners to reject for travel expenses. You also have to look for Toronto, Montreal, Buffalo and possibly even Minneapolis (perhaps at the suggestion of Stoneham), dark horse candidates include Indianapolis, Atlanta and Denver. Denver had just renovated Mile High Stadium (Then known as Bears Stadium) in hopes of luring a team or getting an expansion team in baseball. Denver was supposed to get a team for the failed Continental League at this time frame too.
3) The A's move almost didn't happen, there was a last minute effort to keep the team in Philly, with Connie Mack beginning to have a change of heart when he was convinced by his money grubbing sons to sell the team. They essentially became a Yankees Farm club on the Major League level in Kansas City. In a scenario where the Browns move it is very, very likely that the AL expands possibly to Kansas City and San Francisco in 1942 or 1943. The NL would want to keep up, so I can see Baltimore and Milwaukee landing those expansion clubs. This rules out the A's moving, for now at least, and are likely to stay in Philly until the death of old man Mack. In a scenario like this, the Whiz Kids are likely never to be formed, and the A's remains Philadelphia's first team, and it is the Phillies that move. Where to I do not know, perhaps Minneapolis? Houston? or maybe the Phllies and Braves become the National League's first teams on the west coast, the Seattle Braves and San Diego Padres?
4) No way, no how Griffith moves his team to Dallas, one of the major reasons he moved it to Minnesota was because there were less black people there than Washington DC. The man was a vicious racist, if there is a team in Minnesota already, he'll move to Toronto or Montreal (he was born in Canada if memory serves me correctly). I agree there will be an expansion club in Washington DC, if for nothing else than to appease the real Washington Senators. But don't expect the team to be too far south. Atlanta is a possible choice for an expansion club.
5) Possibly, one reason the Kingdome wasn't built sooner was because the Seattle Pilots owner was broke and was threatening to relocate or sell the team to out of town interests, if there is a stable team located in the area, it isn't too much of a stretch for the Kingdome to be built a decade sooner.
6) All of this depends on how the cards drop, it will look very different thats for sure, but a 71 expansion club will almost certainly find its way to Dallas.
7) Are you suggesting interleague play at this time frame? Baseball traditionalists nearly had a shit fit when it started in the 1990's, and in the late 1960's when they were stronger that they nearly had a stroke when they introduced divisional play.
8) Very likely, though I think an expansion team in the mid 1990's would make more since.