The 26-year-old is not formally suspended by club or governing body but could miss the rest of the season during the investigation, with his participation at Euro 2016 also now in doubt.
"It's a nightmare for the player," Carragher told Sky Sports. "I think he's let himself and the club down badly, because he's become a really important player.
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"There's serious questions to be asked of him.
"He's had a great campaign. He's one of the leaders of the team now, certainly within that run, thinking about how well he played in the Manchester United and Borussia Dortmund games.
"It's now how Liverpool fill that void.
"For a professional footballer, as in any sport, it's not acceptable."