Bay's wife, Kristen, delivered Evelyn Jane at roughly 6:45 ET on Tuesday night. Bay arrived roughly 45 minutes before the birth.
Bay will likely rejoin the Red Sox on Friday in Toronto, where they open a three-game series against the Blue Jays.
It doesn't seem to make sense for Bay to fly all the way back to St. Petersburg for the finale in this three-game series against the Rays, which will take place on Wednesday night.
Red Sox manager Terry Francona had talked at length with Bay over the past few days, and there was some hope that Kristen could perhaps be induced on Thursday, when the Red Sox have an off-day before their weekend series in Toronto.
But it was not to be.
"She just wouldn't cooperate," quipped Francona before Tuesday's game. "That baby is coming pretty soon."
Without Bay, the Red Sox went with an alignment of Jacoby Ellsbury in left, Coco Crisp in center and Mark Kotsay in right.
Kotsay hit sixth, which is where Bay has been making the bulk of his starts of late.
Ian Browne is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.