Minnesota placed Young on the family medical emergency list, which allows players to attend to personal issues for three to seven days.
Twins manager Ron Gardenhire said Young found out after Thursday's victory over Detroit in Minneapolis, as the team prepared to head to New York.
"I missed him. He came and got his stuff at the airport," Gardenhire said. "He found out after the game and took off. He grabbed his luggage and was heading out."
Twins general manager Bill Smith has been in touch with Young's agent, according to Gardenhire.
"A couple of players have texted Delmon. He's hanging in there the best he can," Gardenhire said.
Young is hitting .277 with one home run and 14 RBIs.
Morales batted .340 before being sent down May 6. He's scheduled to be in uniform Friday against the Yankees in New York.