"Very difficult organizational decision," White Sox manager Tony La Russa said. "To have a veteran sit around, not playing, goes against why he was brought here. It's just the reality of the some of the development of our young guys more than anything."
Outfielder Adam Engel, who has also been slowed by hamstring issues this season, was reinstated from the 10-day injured list.
Eaton, 32, was in his second stint with the team after signing a one-year, $8 million deal which included a team option for 2022. He was batting .201/.298/.344/.642 with five homers, 28 RBI and 33 runs scored over 58 games.
"We can now see realistically Eloy [Jimenez] will be back and Luis [Robert]," La Russa said. "Playing time is at a priority now. Too many good players. Good problem to have."
Jimenez (pectoral) and Robert (hip flexor) are both performing baseball activities and should be back sometime in the second half, while Sheets has a .984 OPS in eight games since making his debut a week ago.
Engel, 29, has played in just 10 games so far.