The Reduced Shakespeare Company was due to kick off its latest UK tour by presenting The Bible: The Complete Word of God (abridged) at the council-run venue on 29 and 30 January.
But earlier this month, calls for the show to be cancelled were made by DUP councillors.
The party's Robert Hill told UTV on Thursday that members of the public had approached representatives asking them to "get it stopped" on the grounds that it was offensive.
He said the council was "willing to take a moral stand" and hit back at those who have criticised the decision by claiming it amounts to censorship of the arts.
"Every film in the theatre is censored - that's why there are age limits on what can be seen and what can't. And where do you stop? There has to be a limit somewhere," Mr Hill said.
If somebody said there's going to be a live sex show at the Mill, would people say: 'Yes, that's okay'?
Robert Hill, DUP
UUP Mayor Fraser Agnew also told UTV that he felt the right decision had been made regarding the controversial play, adding that a professional facilitator had been brought in to resolve the issue.
"There were a lot of people concerned about the nature of this play, that it was anti-Christian - and we have established indeed it was anti-Christian," he said.
"I believe there's got to be some form of censorship, otherwise you'll have all sorts of things happening."
Mr Agnew added: "I would hope that we'll (Newtownabbey Borough Council) come out of it in a very sensible light - we've resolved a very difficult situation in a magnanimous way."
On first hearing of the row earlier this month, the show's producer Davey Naylor told UTV that the production had toured extensively around the world over the past 20 years.
"The company has even performed the show at the Jerusalem Festival in Israel without objection," he said, calling for people to come and see it before branding it unholy.
But on Thursday, the Theatre at the Mill confirmed that the two planned performances had been cancelled and that refunds would be available from the Box Office.
It is an expensive u-turn, with the cost to the council estimated to be in the region of nearly £4,000.
Mr Naylor expressed disappointment and sadness at the decision, insisting that the show is a celebration of the Bible and that it is never mocked.
Playwright Austin Tichenor, co-author of the script and one of the managing partners of the Reduced Shakespeare Company, added: "Our show is a celebration of all the great stories of the Bible.
"The theatre is our temple and we're disappointed that people of all faiths will now not be able to gather at the Theatre at the Mill and lift their voices in communal laughter.
"We find it quite staggering that this type of censorship still appears to flourish in the UK and would like to apologise to all the audience members who bought tickets and are therefore unable to see the show.
"We'd also like to thank everyone for the hundreds of messages of support we have received. It's a shame these voices are drowned out by the few dissenting, uninformed ones."
The council explained that the Artistic Council made the decision during an 'away day' on Wednesday to plan and develop a strategy for the future arts programme.
"Subsequently, the chair of the council's Artistic Board, Councillor Billy Webb, has written to the Council's Chief Executive to confirm that, with great regret, the Board has taken the decision to cancel next week's production," a statement from the council said.
"In taking this decision, the Board wishes to confirm its commitment to deliver on the agreed council's Artistic Policy 'to deliver the highest quality performing arts programme, offering a diverse, socially relevant and enriching experience to as many citizens as possible'."
I have not heard the like since the Life of Brian was banned in Belfast 25 years ago in 1979.
Gerry O'Reilly, Sinn Féin
But Sinn Féin's Gerry O'Reilly is among those angered by the handling of the situation, over concerns regarding censorship.
"This is clearly an example of certain councillors forcing their religious views onto everyone else in the constituency," he said.
"What the councillors are basically saying is that they can dictate what type of dramas people can view. This is censorship at its worse."
The show at the centre of the row is billed as "an affectionate, irreverent roller coaster ride from fig leaves to final judgement".
According to the promoters, it tackles the great theological questions like "Did Adam and Eve have navels? Did Moses really look like Charlton Heston?"
They say: "Whether you are Catholic or Protestant, Muslim or Jew, Atheist or Jedi, you will be tickled by the RSC's romp through old time religion."