"Woke up not feeling good," Rays manager Kevin Cash said. "Opted to keep him away. Certainly will test and all that."
Tampa Bay also lost starting pitcher Chris Archer and first baseman Ji-Man Choi during the first two innings of the victory. The AL East-leading Rays started the day with 16 players, including 15 pitchers, on the IL.
The 41-year-old Cruz, acquired from Minnesota last month, is hitting .198 with seven homers and 18 RBIs in 24 games with Tampa Bay. The slugger has a .270 batting average with 26 homers and 68 RBIs for the season.
Cruz is 42nd on the career home run list with 443.
Archer was pulled because of left hip tightness in his first start since since April 10, when he left a game against the Yankees due to right lateral forearm tightness.
He allowed one hit, struck out four and walked one in two innings against the White Sox.
"With the little bit of rest that I've had, the treatment that we've done, I feel a lot better," said Archer, who doesn't believe the injury is serious.
Choi doubled in the first and scored on Austin Meadows' single, but left after the inning with left hamstring tightness.
"He said he felt it when he hit the double," Cash said. "Getting to second base felt it grab him and then probably just continued to tighten up."
The Rays recalled right-hander Chris Mazza from Triple-A Durham to take Cruz's place on the roster.