"This is a game where you have a rivalry situation and, unfortunately you have the worst segment from a very small segment of both fan bases that come and brings about this type of event," York said Monday at a news conference at Candlestick Park. "It's our belief that we should recommend to the NFL that this game is at least postponed for some period of time."
York later said, "It's unfortunate."
The San Jose Mercury News reported that the league and police strongly support ending the traditional meeting between the teams. The newspaper cited a high-ranking NFL source as saying that there is no way the game will be played next year or the year after.
York said the team would even turn away season ticket holders at the gate if officers think they are ready to start trouble.
"This game was like no other that I can remember, and I've been a Niner fan my whole life," Police Chief Greg Suhr said. "Nobody could have been prepared for what happened on Saturday night."
Suhr, Mayor Ed Lee and 49ers team officials said DUI checkpoints will be near the stadium after games as authorities will strictly forbid alcohol consumption at that time.
They also plan to make police and security more visible inside and outside the stadium and urged fans to be more accountable for their actions.
"To those of you who decide to come to our games, and it really doesn't matter what jersey you may be wearing, or what hat you may wear, or what team you may support, your behavior on Saturday night is not welcome," said Jim Mercurio, the 49ers' vice president of stadium operations and security. "Don't come here. You're not welcome."
Earlier Monday, Lee said he was horrified as he watched violent fan confrontations at the game. Lee attended Saturday's game with Oakland Mayor Jean Quan, and both witnessed the brawling firsthand as spectators.
"They were just constantly wailing at each other without regard to who was there," Lee said of the fans. "This is a family outing, for residents and visitors and people who want to see the game, not for people to look for people they don't like, then saying bad words, then getting into it."
Two men who were initially listed as seriously injured in the violence were upgraded to fair condition on Monday.
One of the victims, a 24-year-old man who reportedly was wearing a T-shirt reading "F--- the Niners," was shot several times in the stomach. Police said he managed to make it to stadium security for help despite the injuries.
The other victim is a 26-year-old man who was beaten unconscious in an upper level stadium restroom during the fourth quarter.
Another shooting victim was treated after receiving superficial facial wounds after the game.
Police did not release the name of any victims. No arrests have been made.
Police were seeking motives in the shootings, including whether the attacks were influenced by emotions involving the annual Battle of the Bay exhibition or possibly gang connections.
The San Francisco Chronicle reported that video footage of fights between fans at the stadium have begun to appear on YouTube.
Police said officers ejected at least 70 people from the stadium on Saturday, compared to about 20 at other games. The figure from Saturday did not include several dozen people ejected by team security, Mercurio said.
There were 90 calls for medical service that day, a figure far higher than at a typical Niners home game, officials said.
"I think some of the fans take it too serious," Gore said. "It's a football game. I don't think they should be fighting and shooting and all that."
NFL spokesman Greg Aiello echoed similar concerns.
"We deplore the activities of a handful of fans at last night's game and pledge our full support to Mayors Lee and Quan and to state and local law enforcement agencies," Aiello said in a statement Sunday. "We are carefully reviewing the events to make sure we have a full understanding of the facts. We will continue to work closely with our clubs and law enforcement agencies to support our fan conduct and stadium security initiatives."
Trask added that the incidents were not acceptable to the Raiders or to any National Football League team and "our thoughts are with all affected."
The Niners said attendance at Saturday's game was about 47,000, far less than the 60,000-seat capacity. Suhr said that there were 10 percent more police officers on hand than usual at the game. That total increased by 30 percent midway through the second quarter as the unruliness unfolded.
The recent violence come nearly five months after San Francisco Giants fan Bryan Stow was severely beaten by two men in Los Angeles Dodgers gear outside Dodger Stadium after the archrivals' season opener.
Two men charged in the beating, Louie Sanchez, 28, and Marvin Norwood, 30, have pleaded not guilty. Stow, 42, a Santa Cruz paramedic, suffered severe brain injuries and remains hospitalized in serious condition.
The Associated Press contributed to this report