Abidal, who was diagnosed with a cancerous liver tumour in 2011 and had a liver transplant the following year, has struggled this season, showing a worrying lack of pace at both club and international level.
The 34-year-old was omitted from Didier Deschamps’s squad to face Netherlands in a friendly earlier this month.
Deschamps has repeatedly said there is no room for off-field dramas in the France team, and stressed players have to be performing for their clubs if they are to be picked for the June 13-July 12 tournament in Brazil.
Monaco coach Claudio Ranieri said leaving the centre back out on Sunday was a sporting decision, and made it clear Abidal, who took part in Les Bleus’ World Cup qualifying campaign, would be asked to explain himself.
“It’s his problem, I know he was at home. He did not stay for the game,” Ranieri said. “If there is a problem I will talk with the club. I want to know why he left.
“He came with us (to the pre-game camp) and he left. He was not on the bench. He left.”
After recovering from a stomach bug, Abidal was in the frame for a place in the squad but Ranieri went with Nicolas Isimat-Mirin, who slotted in while Abidal was not fit.
“He had had a good week, he had trained every day, but I preferred another player.”
Ranieri now has to decide whether Abidal will be part of the squad to face Ligue 2 side RC Lens in a French Cup quarter-final on Wednesday.
“I will see how it goes in the training session and I will then make a decision,” the Italian said.
Monaco’s Ligue 1 title challenge all but ended when they drew 1-1 with at home to Lille on Sunday, leaving them 10 points adrift of leaders Paris St Germain.
(Reporting by Julien Pretot; Editing by Peter Rutherford)