Green, who is Black, was playing for Greuther Fürth against host Hallescher FC in the first round of the competition.
On the Fürth club website, Green said a spectator racially abused him during the game.
Fürth coach Alex Zorniger said the abuse occurred repeatedly and called for other fans to intervene in cases of racist abuse.
"The stadium was 95% full. There were enough people there who could have intervened," Zorniger said in comments reported by Germán news agency dpa.
The dpa agency also reported that Halle club spokeswoman Lisa Schöppe issued an apology on behalf of the club.
Green, who began his career at Bayern Munich, has played for Fürth in Germany since 2017 and also played 15 times for the USMNT from 2014 through 2018.
He was selected for the 2014 World Cup in Brazil aged just 19, and scored in the United States' round-of-16 defeat to Belgium to become the youngest American to score at a World Cup.
Second-tier Fürth beat third-tier Halle 1-0.