German television broadcasters aired the grainy 90-second video clip at the weekend.
The instructor says in German while a recruit lying on the ground starts firing his gun: "You're in the Bronx, a black van pulls up in front of you and three African-Americans get out and start really insulting your mother ... now do something!"
The soldier fires his machine gun and yells an obscenity several times in English between bursts. The instructor then tells the soldier to curse even more loudly.
In New York, Adolfo Carrion, Bronx borough president, has demanded an apology.
"The German government obviously has work to do to correct something that is insidious," he said.
"Clearly these folks don't know anything about African-Americans or the Bronx."
The German army has had to deal with a number of embarrassing scandals in recent years that damaged its image.
These include the publication of pictures of German soldiers in Afghanistan who took pictures of each other holding skulls and allegations that torture has been used during training.
The investigation is likely to continue for several weeks.