Rodriguez left the club after Sunday's game to have his strained right quadriceps tested in New York, then flew to Miami, where his wife, Cynthia, gave birth to a 7-pound, 2-ounce baby girl on Monday evening.
"I'd be shocked if he played tomorrow," Girardi said on Wednesday. "If he comes in and says he's ready, believe me, I'm putting his name in there. There's no hesitation where I'm putting it, either. But we've got to make sure."
Rodriguez injured himself legging out a fielder's choice on Sunday in Baltimore and is listed as day-to-day. Girardi said that he expects Rodriguez to test the leg upon returning to the club but anticipates the slugger will be somewhat cautious with it. Shortstop Derek Jeter recently missed six games with a similar injury.
"You tell them to be smart, and inevitably, they're on first base and a guy hits the ball in the gap," Girardi said. "It always seems to happen to the guy you tell to be careful. We've just got to make sure he doesn't push it too much too early. He's going to want to play -- you know Alex."
In other injury updates, Yankees first baseman Jason Giambi was back in the lineup on Wednesday after he took a ground ball off his left thumb in the seventh inning of Tuesday's 9-5 victory over the White Sox.
Giambi batted in the eighth inning and was removed for defensive purposes that Girardi called precautionary. Giambi left U.S. Cellular Field with his hand wrapped in a soft bandage.
"It was just a little swollen," Giambi said on Tuesday. "[Girardi] just wanted me to take care of it. Hopefully, it should be fine."
Giambi was 1-for-4 with a solo home run on Tuesday and is batting .120 this season. With just four hits in his last 23 at-bats, Giambi said that his low batting average does not concern him at this time.
"I'm more worried about taking good at-bats and hitting the ball hard," Giambi said. "That's the only thing you can look at and concentrate on."
Bryan Hoch is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.Less