Cleveland called up right-handed pitcher Tom Mastny from Triple-A Buffalo to take Elarton's roster spot.
Elarton flew with the team into Chicago on Sunday night and told Wedge on Monday morning that he needed to leave the team.
"He has some personal issues that he has to take care of," Wedge said, "so he headed out of here today."
Wedge said Elarton would be gone "indefinitely."
Elarton, signed to a minor league deal in February, had his contract purchased from Triple-A Buffalo on May 24. He made eight appearances for the Indians, going 0-1 with a 3.52 ERA.
The 6-foot-8 right-hander, who first came up as a starting pitcher with Houston in 1998, has made 232 career appearances, including 170 starts. He is 56-61 with a 5.29 ERA in his big league career. The 32-year-old Elarton has also pitched for the Rockies and Royals.
Mastny was 1-0 with a 1.71 ERA in the minors this season and is 7-4 with a 5.49 ERA in three seasons in the majors, all with Cleveland. He's 0-1 with a 21.00 ERA in three appearances with the Indians this season.