SDLP Councillor Gerry Mullan who called for a recorded vote on the matter is appealing to the 17 councillors who voted against the project to "pull back from the brink".
Mr Mullun said: "It is very difficult to see how any court could rule in their favour given that the shortfall for this project is £300,000 which would be paid out of Community Landfill Tax and if that doesn't happen, due to unionist objections, the ratepayers will bear the cost of renovation of the existing facilities which could be in the region of £1million.
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"This is exactly what happened on Craigavon Council many years ago when unionists refused to accommodate the needs of the GAA in that area. The Dungiven situation is very similar and when you consider the needs of people with disabilities it is more serious. The present facilities are hopelessly inadequate and the Causeway Coast and Glens Council is blocking its replacement even though not a single penny of ratepayers money is being asked for."
Sinn Féin MLA Cathal Ó hOisín also said the decision by unionists "beggars belief".
He said: "I cannot think of any other council who would reject a £2.5 million funding package to build a much-needed facility.
"The Culture Minister has committed the vast majority of the funding for the project with the remainder coming from the Landfill Community Fund.
"In an even more ridiculous move; this council is also proposing incurring costs of £800,000 to carry out refurbishments on the existing facility, which is long past its sell-by date instead of building a new one.
"Essentially, the council is advocating spending £800,000 to block a £2.5 million pounds investment. This is absolutely ridiculous."
(CD/LM)
