Two senior columnists at the paper, Fintan O'Toole and John Waters, called for an apology as readers aired their disgust on radio.
Mr Waters said it could be a resigning matter for editor Geraldine Kennedy.
Mr O'Toole told RTÉ radio he was disgusted and embarrassed by the article, which he believed would have been rejected by the worst Irish tabloids.
He said he couldn't believe the article appeared in the Irish Times.
"If he [Kevin Myers] is not happy he should come out and say why it is OK to attack children, you should not attack people who cannot answer back," Mr O'Toole added.
Mr Waters told Newstalk 106: "I think this is potentially a resigning issue for the editor unless this is addressed adequately, today, then the editor should be removed in my view."
Irish Times journalist Kitty Holland also called on Mr Myers to apologise, saying she was the daughter of a single mother, the late Mary Holland.
"I think there should be an apology because his column went out with a supplement for schools that was read by thousands of illegitimate children," she added.
Irish Times editor Geraldine Kennedy said the issue would be addressed in today's edition and Mr Myers refused to comment. Both Mr Myers and Ms Kennedy declined several invitations to comment on radio yesterday.
There was a huge public outcry in response to Mr Myers's claims that many girls "consciously embark on a career of mothering bastards because it seems a good way of getting money and accommodation from the State".
Irish Times trust governor David Begg said he was shocked by Mr Myers's comments and he believed he should apologise.
"Mr Myers went over the top in his comments and I will be asking the Irish Times Trust to discuss the matter at our next meeting," the Irish Congress of Trade Unions general secretary said.
Archbishop of Dublin Diarmuid Martin also condemned Mr Myers's use of "hate words".
"Lone parents love their children they work twice as hard as others," the archbishop added.
Single parents organisation, One Family, said the term 'bastards' was incredibly offensive.
"I would characterise the article as 'journalistic thuggery' and it also had a litany of factual inaccuracies which we regret," One Family spokeswoman Karen Kiernan said.
NUJ secretary general Seamus Dooley said he also found Mr Myers's comments deeply offensive and he criticised the Irish Times columnist for not going on RTÉ to defend it.
"The term bastard has no place in a column.
"His analysis was simplistic and conflicts with the socially progressive editorial stance of the Irish Times the paper must address the public reaction," Mr Dooley said.
But former Sunday Independent columnist Mary Ellen Synon, who stopped writing for that newspaper after writing a critical article on disabled athletes, defended Mr Myers.
"Kevin expressed an opinion, I gather his facts are right, it's an opinion, get over it, what's the problem?" she told RTÉ.