"The Argentinian defender experienced the setback in our Premier League game against Arsenal earlier this month," the club said in a statement.
"Martinez bravely continued playing for the following two games against Brighton & Hove Albion and Bayern Munich.
"But it has now been determined that he needs a period on the sidelines for recovery and rehabilitation. Assessment is ongoing to decide next steps."
Sergio Reguilon (26) will also be absent from Saturday's Premier League game against Crystal Palace with a minor injury.
Ten Hag said that midfielders Scott McTominay and Christian Eriksen would return to the side having missed the League Cup victory through illness. The Dutchman also expected a tougher encounter against Palace this time at Old Trafford.
"Definitely, both sides will be different but we got belief from it (Tuesday's win). I know the opponent is looking for revenge and that will be their view," he said.
United also confirmed that Antony will return to training after allegations of abuse, and Ten Hag is willing to consider the Brazilian winger's inclusion in his squad.
"Everything has been made clear in the statement. No, I don't think it (Antony's return) is a distraction. We will focus on the games. He will do as well. I think we explained it in the statement, so I will refer to it," Ten Hag said.
"On Saturday, he will be back in training, so tomorrow. I have to see, but I think yeah (he is fit)."
The Dutchman also addressed the issue of injuries in an ever-expanding fixture list. United face Galatasaray in the Champions League three days after the Palace game.
"We already expanded the squad this season because we made the reason clear. We had a World Cup in the middle of the (last) season, we had a longer season," he said.
"Every time the schedule is expanding, the load on the players is too much. Many colleagues of mine have pointed to that and I have as well. But it keeps going."
Manchester United are in ninth place in the league standings, one point ahead of Crystal Palace who are 10th.
