The chair umpire was forced to intervene when the two players clashed during their first-round meeting, which Harrison won 6-3 7-6(4).
After the match, Young took to social media and made the allegations.
"I'm shocked and disappointed to hear you [Harrison] tell me how you really feel about me as a black tennis player," he wrote on Twitter.
Harrison has denied the claims and insisted that video and audio footage from the game will "100% clear [him]", but the Association of Tennis Professionals (ATP) has launched an investigation into the incident, reports PA.
Young is said to have refused to shake hands with Harrison at the end of their New York Open meeting.