Last season, Derek Jeter had a similar injury that cost the Yankees' captain three weeks on the disabled list, halting his pursuit of the 3,000-hit milestone.
"You can't feel sorry for yourself," Jeter said. "No one's going to feel sorry for us."
Because rosters will expand to 40 players on Sept. 1, the Yankees can maneuver without placing Teixeira on the 15-day disabled list, as they did with Jeter.
Instead, they figure to fill in at first base using an arrangement of Nick Swisher, Eric Chavez, Casey McGehee and newcomer Steve Pearce, who was acquired from the Astros on Monday.
"We're going to have to step it up," Swisher said. "We've been dealing with stuff like this all year. As bad as it is to say, I guess the show goes on, you know?
"Tex is one of those guys; he's been huge for us in the clubhouse, on the field, so anytime you miss a guy like that, we're going to have to do some work."
Teixeira said he felt his calf grab after a foul ball during his at-bat, which resulted in a walk. Teixeira stepped out of the box at one point and flexed the calf, believing he could play through the tightness.
"I thought I could run through it, finish the at-bat; I drew the walk," he said. "Instead of loosening up as I was running, it tightened up more."
Teixeira grimaced as he chugged around the bases on Russell Martin's RBI single, then disappeared immediately down the clubhouse runway. It could be a while before his next appearance in a lineup.
"I don't see him playing in this series," manager Joe Girardi said. "I'm not even sure I see him playing in the next series, by the way I saw him run. [Sept. 1] is Saturday, so I don't know if that means anything, but we'll see. ... Calves can be really tricky. We've seen that from guys. I'm concerned."
Teixeira is batting .255 with 23 home runs and 81 RBIs in 119 games this year. Teixeira has also been managing a sore left wrist he said he might have to deal with all season.
"It's very frustrating," Teixeira said. "We've had our share of bad luck, as most teams in baseball have. We've just got to fight through it. We've had guys step up all year for us, so whether it's a week, two weeks, whatever it may be -- we'll have guys step up."