Abidal, who will turn 34 in July, has one match left with the Spanish league champions, but said he would have liked more.
"Football for me is a passion since I was a child," Abidal said at Camp Nou. "I don't see myself quitting now. I have made all this effort to come back to play football. My health is better now. When my body tells me to stop I will hang up the boots.
"I would have liked to keep on playing here, but the club sees it differently and I have to respect that."
Barcelona president Sandro Rosell said the door is open for Abidal to return and join its technical staff when he finishes his playing career.
Abidal thanked his family, teammates, coaches, and the team doctors and physical therapists that helped him through his medical issues. He battled back to play for Barcelona in April after having a liver transplant last year following a bout with cancer in 2011.
Even though he was obviously disappointed that Barcelona had not offered to extend his contract, Abidal said that he had spent the "most intense years of his life" at the club.
"I am going but I am sure that I will be back," he said. "I have a very interesting offer from the club that I will have to think about."
Abidal was visibly moved by the moment and showed some tears during a tribute video the club dedicated to him. He then embraced and kissed the cheek of each of his teammates, coaches, doctors, staff and club board members before leaving the press hall.
Abidal said he wants to keep playing as long as his health allows, but that he had not received any offers from other clubs yet and that he would wait until he finishes his contract until considering his next move along with his wife and children.
Neither Rosell nor sports director Andoni Zubizarreta offered a clear explanation on why Abidal had not been offered a new contract. Zubizarreta even said that Abidal's recovery following the transplant of a piece of his cousin's liver had exceeded expectations. But with Barcelona's back line aging, it appears the club may be looking to overhaul its defence this off-season.
"Eric has been with us for six years, and it goes without saying that he has become a very important person for us," Rosell said. "He has won our hearts. He showed us the will to overcome any type in difficulty.
"Today we are not here to say goodbye, or until later, but rather until he wants to return. The door is open for when he wants to become technical director of all our football academies. We will wait for him. When he returns, if he wants to help us in another way, then we will consider that also."
Abidal formed a key part in Barcelona's most successful run in club history, winning 15 titles in his six seasons.
The former France defender joined Barcelona from Lyon for the 2007-08 season. After going through a transition period adapting to Barcelona's passing style, which requires its defenders to handle the ball under pressure, he slowly but surely became one of the team's first-choice players and also a fan favourite.
Starting as a left back, Abidal ended his stint with Barcelona at centre back and as one of its most reliable defenders and veteran leaders.
"A great player, teammate and above all an example to follow! I wish you all the best my friend!" Barcelona midfielder Cesc Fabregas, who was present with the rest of the team, wrote on Twitter.
Besides two Champions League and four Spanish league titles, Abidal also helped the Catalan club win two Copa del Rey titles, two Club World Cups, two European Super Cups, and three Spanish Super Cups.
Abidal's last game will be on Saturday when Barcelona concludes the league against Malaga with the chance to equal a record of 100 points with a victory.
"The most important thing right now is my last game," Abidal said. "Let's see if my coach plays me and I can have some minutes."