Smith was placed on the 10-day disabled list on Tuesday.
"I don't agree with it," Cora said when he met the media Wednesday at Fenway Park before the team's series finale against Oakland. "On a daily basis, we talk to pitchers and see how they feel. "If they don't feel that they can pitch that day, we stay away from them.
"It caught me by surprise. If he felt that way, he should have talked to us."
Smith, a 28-year-old right-hander, was injured when he threw his glove during a tantrum in the dugout after leaving Monday night's 6-5 loss to Oakland. He entered with the Red Sox trailing 5-4 in the eighth, allowed Khris Davis' leadoff home run, then retired three straight batters.
After he was placed on the DL, Smith opened up.
"I think fatigue played a factor with my shoulder, and my shoulder just couldn't handle it," he said. "Think my shoulder's tired in general. Yeah, it's just from pitching. I've thrown a lot lately, and I think my arm was just tired."
Smith said he tossed his glove regularly.
Cora said he hadn't had a chance to talk to Smith yet, but that he will.
"We ask everybody, all the relievers [how they feel]," Cora said. "We talk to them on a daily basis: 'Where you guys at?' We have information that's very important to our decision-making. There have been some days that he felt like he wasn't available."
Smith was 1-1 with a 3.77 ERA in 18 games before he went on the DL. He was acquired from Seattle before the 2016 season to be a setup man for closer Craig Kimbrel but has been limited to 29 games for the Red Sox. He had Tommy John surgery two years ago and returned to appear in eight games last September.
Cora said the club had no update on Smith's recent injury, and that there would be more tests.