The 34-year-old former France international managed only two appearances for Bani Yas after joining the club in August, with a long-standing thigh injury forcing him on to the sidelines.
Trezeguet had grown frustrated with the niggling problem and had asked Bani Yas last month to cancel his one-year contract, believed to be worth €1.6 million (Dh7.8m).
Bani Yas had said they wanted to hang on to his services, with Trezeguet's agent Antonio Caliendo flying into Abu Dhabi last week to discuss the situation with the club.
However, Calderon, who was appointed Bani Yas coach 10 days ago, said on Sunday he would allow Trezeguet to leave as he didn't want a squad member to be somewhere he didn't want to be.
"David is a good person and I have a lot of respect for him because he was a big player," Calderon told reporters.
"But I let him go. I told him to go. David did not want to be here and I do not want unhappy players at the club. It was best for everyone that he has gone."
Bani Yas spokesperson Musaad Al Harthi said in a press statement: "David did not want to change his mind and we have to respect that. We wish him every success in the future and it is a shame that this did not work out."
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