Young was suspended for seven days by Major League Baseball three days after he was arrested in New York.
Young allegedly confronted a group of men outside a hotel two weeks ago. The altercation included the slurs and a physical attack, authorities said.
Police said they took Young to a hospital because of acute intoxication and he was jailed on a charge of misdemeanor aggravated harassment.
MLB officials suspended Young effective April 27.
"Those associated with our game should meet the responsibilities and standards that stem from our game's stature as a social institution," MLB Commissioner Bud Selig said at the time.
"An incident like this cannot and will not be tolerated. I understand that Mr. Young is regretful, and it is my expectation that he will learn from this unfortunate episode."
Young is batting .242 in 18 games this season with one home run and five runs batted in. To make room for Young on the roster the Tigers put the contract of designated hitter Brad Eldred up for assignment. Eldred was 3-of-16 batting while with Detroit.