Rosario, hurt running out a grounder in the second inning of Monday night's 9-8 loss at Tampa Bay, joined a list of key players out that also includes starting pitchers Shane Bieber, Aaron Civale and Zach Plesac.
Rosario also left last Thursday's game against Houston with right abdominal tightness, one day after fouling a pitch off his left cheek against Detroit. He sat out Friday's game against the Astros.
"We think it's mild," Cleveland president Chris Antonetti said. "That's our best assessment at this point. We just felt with the continued lingering soreness in there it made sense to make the IL move, especially with the All-Star break and you're not playing games for four days. It just seemed to make sense that now was the right time to give him a bit of a rest."
Plesac is scheduled to return and start Thursday night's game against Kansas City after being out with a broken thumb on his pitching hand. The right-hander, 4-3 with a 4.14 ERA in 10 starts, was hurt while yanking off his undershirt following a rough outing in late May.
All-Star Shane Bieber, placed on the 10-day injured list June 14 with a right shoulder strain, will be examined this weekend to determine if he is ready to start a throwing program. The 2020 AL Cy Young Award winner is 7-4 with a 3.28 ERA over 14 starts.
Civale, out since June 22 with a right third finger sprain, is playing light catch.
Antonetti said there is no definitive timetable for when Bieber and Civale will be back.
"It's safe to say we're still weeks away," Antonetti said.
Bieber, Plesac and Civale are combined 21-9 this season for the Indians, who started Wednesday at 42-40.
Cleveland, which entered Wednesday's doubleheader with the Rays on a seven-game skid, made a number of moves that included clearing a roster spot by transferring outfielder Josh Naylor from the 15- to the 60-day IL.
Naylor sustained season-ending multiple leg fractures, a broken ankle and torn ligaments in a collision with a teammate during a game at Minnesota on June 27.
"We have a whole offseason in front us and we'll learn more over the course of the next few weeks as he really starts to get into the rehab process," Antonetti said.
Naylor and rookie second baseman Ernie Clement ran into each other as they converged on a fly ball in short right field. Naylor's foot got trapped underneath him after he slammed into Clement. Naylor was sent sprawling and his lower right leg was bent back awkwardly.
Right-hander DJ Johnson had his contract purchased from Triple-A Columbus, while left-hander Logan Allen was optioned to the minor league club. Outfielder Daniel Johnson was recalled from Columbus and infielder Owen Miller was added as the 27th player.