Councillor Ross Brown backed the 'Stop the Drill' campaigners at the site over concerns the drill may affect the drinking water in the area.
He said he was "furious" that the drinking water supply for tens of thousands of people is under threat.
However, NI Water has confirmed there is no danger to the water supply.
Cllr Brown said: "There has been a lack of transparency in the decision making process that has allowed the drilling to start. A strategic environmental assessment should have been carried out.
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"There is no community consent for this activity. I am astounded that it appears that the rights given to Infratrata were done so by default as the Council did not make an assessment on the decision.
"I would like to pay tribute to the campaigners who have kept this issue in the public domain. The judicial review is also an important next step in the process to have the drilling stopped, but it saddens me that the precautionary principle is not applied and that the drilling will continue while the court proceedings are ongoing.
"The Green Party will continue to support the campaign to stop the oil drilling and will take this issue up again in the Assembly once Ministers have been appointed."
Sinn Féin MLA Oliver McMullan said the issue should be a priority for the new Executive.
He said: "There are genuine concerns in the local community around Woodburn Forest about the potential impact this drilling could have on the environment.
"I have visited the site and met with members of the campaign group and with representatives of the company behind the drilling, Infrastrata.
"I am once again calling on Infrastrata to meet with local residents and campaigners to discuss the concerns around this development."
(CD/JP)
