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.
"Discussion with the vice president and the coach. Everything back in order. Apologies made to the team," Abidal, who was omitted from Didier Deschamps's France squad to face Netherlands in a friendly earlier this month, wrote on his Twitter feed on Monday.
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.
"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."
On Monday, Ranieri put the incident to bed, telling a news conference: "It's over, there is no problem.
"It was important that he stayed with the squad because he is the captain. But I won't hold it against him."
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 explained.
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.
They take on Ligue 2 side RC Lens in a French Cup quarter-final on Wednesday.
- Sports & Recreation
- Didier Deschamps