Kuroda is scheduled for an MRI on Wednesday morning.
"He's experienced this on and off for a while, but we just found out about it," Torre said. "He was ready to pitch [Wednesday], but we decided to send him home. We'll know more by the time we got home, probably before we get home."
Kuroda experienced some discomfort while throwing a bullpen session on Sunday, Torre said. When Kuroda informed Torre of the soreness on Tuesday, the manager opted for the safest option.
Derek Lowe will start in Kuroda's place on Wednesday. Triple-A Las Vegas callup Eric Stults will pitch Thursday, while Clayton Kershaw will take the mound on Friday against the Indians.
Kuroda is 3-6 through 14 starts with the Dodgers this season with one complete game. He lasted just 2 1/3 innings and gave up six runs on five hits in his last start on Thursday at San Diego.
Torre would like to have known about Kuroda's condition earlier.
"When he's been able to pitch, he's pitched," Torre said. "His numbers have been good. But I would have liked to have been told sooner because I'm concerned about the health and welfare of the pitcher.
"Knowing the discipline that he's pitched under, he felt a great deal of responsibility. This was tough for him to talk about based on the fact that he's not going to be able to take the ball tomorrow, which he was willing to do.
"I told him there's no win that's more important than his career."
Brandon Harris is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.