"It will take about three weeks to heal, so he's obviously done for the year," interim manager Daren Brown said.
Carp, who spent most of the season with Triple-A Tacoma, was injured while running out a double in Saturday's game against the Rays at Tropicana Field.
He had been recalled from Tacoma three days earlier.
Meanwhile, designated hitter Russell Branyan was having more tests done on his lower back, and although he has not been ruled out for the season, it appears unlikely that he would play in the season-ending four-game series against the Athletics at Safeco Field, which begins on Thursday night.
Right-handed reliever David Aardsma also could be done for the season.
He was supposed to play catch prior to Tuesday night's game against the Rangers, but that was canceled after he felt some discomfort in his left oblique.
Brown said right-hander Brandon League would handle the bulk of the closer duties, and if he needs a day off, rookie right-hander Dan Cortes would be the leading candidate to protect the lead.