Kuroda was sent to Los Angeles for an MRI on Thursday while the team was playing the Padres. No structural damage was found.
Conte termed the strain "mild," and said Kuroda could start throwing again in four to five days if the tightness disappears.
"It isn't pain, it's tightness in a very specific area where we worry about for pitchers, on the opposite side," Conte said. "The key is we think it's mild, but it can be serious and take six to eight weeks. We think we caught this early."
Kuroda was disabled retroactive to Tuesday, so he's eligible to be reactivated April 22.
"I pitched Opening Day and felt really good, so to not pitch really disappoints me," Kuroda said. "I was going to start, but I have to respect Joe Torre's opinion and listen to the trainers and do the best I can on the rehab program.
"I've calculated so I can be ready [on April 22]. I've put a lot of burden on the pitching staff, so I'll try my best to recover as fully as possible and help the team as much as I can."
Torre said he was "hopeful" he'd get his Opening Day starter back that quickly.
"Pitchers are different than players," Torre said. "Hopefully, he'll improve as it goes along. It doesn't seem all that serious, but serious enough to DL him. He was very disappointed. I talked to him twice today. He wanted to pitch."
Kuroda pitched effectively Monday in San Diego, allowing one run on four hits over 5 1/3 innings. But he ran into trouble in the sixth inning and loaded the bases with one out before being relieved by Cory Wade, who put out the rally.
The 34-year-old Kuroda made 31 starts in 2008, his first season with the Dodgers after 11 seasons pitching in Japan. He spent one 17-day stint on the disabled list with shoulder irritation.
Losing his Opening Day starter reminded Torre of losing David Cone to an aneurysm in 1996, when he was the Yankees' manager.
"The No. 1 guy went down, and Doc Gooden throws a no-hitter," Torre said. "You hope you find good things out of this. The main thing is his health. We've got a lot of games left. He just felt he let people down."