Despite a Department for Infrastructure civil servant approving plans for the facility in 2017, the High Court blocked the project and ruled planning approval should be made by a minister and not civil servants.
SDLP Deputy Leader, Nichola Mallon MLA, welcomed the Department for Infrastructure's decision not to further appeal Court rulings.
"A huge amount of public money has already been wasted trying to defend the indefensible," she said.
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"This incinerator should never have been approved in the first place. It is right that the Department for Infrastructure has decided not to pursue this case further.
"I want to pay tribute to the No Arc 21 Committee and local residents in North Belfast and South Antrim who have tirelessly fought against this unneeded, unsustainable and unviable facility.
"As these Court rulings have made clear, civil servants have no legal authority to take these decisions. The question for the DUP and Sinn Fein is how long are they going to continue to deny the people of Northern Ireland a Government that will take decisions in their interests and end this dangerous drift."
Sinn Féin MLA Declan Kearney also welcome the decision.
He said: "The Department of Infrastructure's decision not to appeal the Court ruling to reject the Hightown Incinerator is the right thing to do.
"This facility should never have been given the green-light in the first place after widespread opposition and concerns over the safety of residents.
"Any move by Department officials to proceed with this project would have run contrary to the Good Friday Agreement in the absence of an Executive."
(LM/MH)
