The 29-year-old suffered a foot problem during a training session with his teammates ahead of the game in Salvador.
Miller had been expected to start for Costa Rica this weekend in place of the suspended Oscar Duarte, but his participation is now under threat.
"The scan showed he had a slightly torn muscle in his left foot," read an official statement from the Costa Rican Football Federation. "He will not be able to train for several days."
Miller's only appearance so far at the World Cup came during his country's goalless draw with England.