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The former England captain played the final 33 minutes but was unable to lift the Galaxy in a game that was goalless until former Manchester United trainee John Thorrington's goal in second-half stoppage time.
Beckham put a long-distance shot over the bar in the 77th minute and his 82nd and 90th minute free kicks sailed dangerously into the box before being cleared.
He was appearing in his second game since returning from a knee injury that kept him out for 10 matches.
Galaxy needed victory to earn a spot in the eight-team playoffs. The Fire were also playing for their post-season lives, but only needed a draw.
The Galaxy entered the match a seven-game unbeaten streak which put them in position to reach the playoffs after a dismal start to the season.
"We've had a good run, but obviously early season is what's affected us more than anything," Beckham told reporters. "We have to win games throughout the season instead of relying on the last five or six games to come back into it. But we've ended the season sort of on a high -- apart from this game -- and we have to be positive about next season."
Beckham said he was not necessarily looking forward to having time off.
"I'm a person that likes to train every day and work every day, and that's the way it's been throughout my career," he said. "For the last 15 years, I've had no longer than three weeks off in the summer. I'm not going to know what to do with myself. I'll be keeping in training and keeping fit no matter what."
Galaxy head coach Frank Yallop said he had no regrets about starting Beckham on the substitutes' bench.
"It would be tough to ask him to (play a full game) with not playing for seven weeks," he said.
"Once he got into the game, he had some good long passes and his free kicks are always dangerous.
