The immigration minister dubbed Samantha Maiden, political editor for New Corp's Sunday papers, a "mad f***ing witch" in a message he accidentally sent her about a column critical of his colleague Jamie Briggs.
The text was reportedly intended for Mr Briggs, who has stepped down after acting inappropriately towards a diplomat while on an overseas trip.
"It's a solid sledge," she told the Nine Network on Monday.
"I sent him a text immediately saying `you know mate, you've sent that mad witch text to the mad witch' and he was more than happy to apologise right away."
Acting Opposition Leader Penny Wong said the Turnbull government is in chaos four days into the new year.
"This isn't a backbench MP - this is a senior cabinet minister," she said.
"It's all very well to talk about increasing the role of women and stronger ministerial standards - today is the day Malcolm Turnbull must put all that talk into action."
Senator Wong called on Mr Turnbull to front the media and explain how Mr Dutton's behaviour is acceptable.
"If he fails to front up, he's sending the clear signal he thinks this behaviour is appropriate."
Ms Maiden said Mr Dutton made the decision to own up, after she chose not to name him when appearing on Sunday television.
She hoped he doesn't end up in the backbench in light of the message mix-up.
"Mr Dutton is a good minister, he's a hard worker, he's made a strong contribution to the government," she said, adding there are more substantive issues in relation to the Briggs affair.
In a statement on Sunday evening Mr Dutton said he was expecting a tough time in the journalist's next column.
"Sam and I have exchanged some robust language over the years so we had a laugh after this and I apologised to her straight away, which she took in good faith," he said.
The Australian newspaper reported that Mr Turnbull had privately described the message as "completely inappropriate".
Education Minister Simon Birmingham said the incident was "unfortunate", but Mr Dutton recognised his error and immediately contacted the journalist to apologise.
"(He) of course has come out and publicly acknowledged that it was him, to avoid there being a witch-hunt or blame being pointed in other directions," Mr Birmingham told ABC radio.
A number of social media users made their views clear using the hashtag DumpDutton.
Others lamented the way the language and behaviour of politicians had fallen to such low standards.
Feminist online campaign group, Destroy the Joint said the language used by the senior federal minister is unacceptable.
Others criticised Minster Dutton for being hypocritical with his condemnation in the past of sexism on White Ribbon Day and sexist comments directed at Labor MPs.
Sunday's text message is the latest gaffe by Minister Dutton, after he made a joke about rising sea levels in the Pacific.
The exchange last September with then prime minister Tony Abbott was recorded off-mic as the pair prepared for a press conference. Pacific Island leaders condemned the comments as vulgar.
Others social media users rallied in defence of Minister Dutton, saying the reaction has been overblown.
