Beckham exchanged words with a man in an England shirt during the Galaxy's 1-1 draw with the Kansas City Wizards.
The 34-year-old, who was fined after a clash with a fan last week, claimed the spectator insulted his wife Victoria.
"He shouldn't actually be wearing that shirt because what he was saying about my wife was a disgrace," he said.
The former England captain was fined $1,000 (£600) by Major League Soccer last Friday following a confrontation with fans during a recent friendly against AC Milan.
Beckham, who missed part of the Major League Soccer season earlier this year when he joined the Italian giants on loan, was booed by a section of the home crowd during the game.
At half-time, he approached one group of fans and appeared to challenge them to meet him on the pitch.
Security staff intervened, and police arrested a fan who left the stands.
Beckham joined Milan on loan because he was keen to satisfy England boss Fabio Capello's demand that he play at the highest level to remain in contention for international selection.
He is England's most capped outfield player, with 112 appearances, and is keen to play on until the 2010 World Cup finals in South Africa.

Source: BBC