The 25-year-old Delmonico, who can play a couple of different positions, got hurt during Thursday night's 5-1 victory against Minnesota. He was scratched from Friday's lineup against Detroit.
Delmonico made his major league debut on Aug. 1. He is batting .307 with six homers and 12 RBIs, reaching safely in 21 of his 22 games.
The White Sox also promoted catcher Rob Brantly from Triple-A Charlotte before Saturday night's game against the Tigers. The 28-year-old Brantly, who signed a minor league deal with Chicago on June 29, is a .225 hitter with five homers and 32 RBIs in 112 career major league games.