Coach Bruce Arena brought on 22-year-old Michael Stephens in place of the experienced midfielder after his side went in at the break 2-0 down, and the 36-year-old left the stadium without speaking to reporters.
Beckham recently signed a two-year extension to his contract - despite speculation linking him with a move to big-spending Paris Saint Germain - and cited former USA national team coach Arena as one of the reasons.
However, the American boss admitted that the former Manchester United man was angry, fuelling speculation that the relationship between the two is becoming strained.
He told reporters: "I’m sure he is not pleased about it. But that’s not the issue in the game.
"The issue in the game was that 11 players from New England soundly outplayed 11 players from the Galaxy.
"I don’t [know] of any players that would be happy about coming out at half-time. I just felt we needed a little bit more.
"We needed to make a change in the midfield. It was purely a coach’s decision. I couldn’t change 11 players. That’s what I told the team at half-time. If I had 11 substitutions, I would have considered that.
"I can’t give you all the answers right now, that’s for sure. It wasn’t good."
Irish international forward Robbie Keane, who signed for the Los Angeles outfit in summer, went on to explain that the team should share responsibility for the defeat.
He said: "We did the exact same as what we have been doing in the last few games, conceding sloppy goals.
"It is stuff you do when you are a kid. We didn’t do it properly, so I guess we have only got ourselves to blame.
"It will be a long week for the boys. It is up to everyone to pick their heads up and be men and get on with it. It wasn’t good enough from any of us."
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