Immigration Minister Peter Dutton dubbed Samantha Maiden, political editor for News Corp's Sunday papers, a "mad f***ing witch" in a message he accidentally sent her about a column she wrote criticising his colleague Jamie Briggs.
The text was reportedly intended for Mr Briggs, who stepped down from the ministry last week over an incident with a female public servant in a Hong Kong bar.
Mr Turnbull admonished Mr Dutton over his text.
"As I advised the minister yesterday his language was clearly inappropriate as he recognised by apologising immediately to Ms Maiden," he said in a statement to the media on Monday.
Acting Opposition Leader Penny Wong says Mr Turnbull needs to explain how the behaviour is consistent with ministerial standards.
"I don't think Australians think it's acceptable and I think this really is a test for Malcolm Turnbull," she told reporters in Adelaide.
After speaking a lot about the importance of respecting women, he needs to put his words into action, she said.
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But Ms Maiden says she doesn't want the minister to end up on the back bench over the incident.
"Mr Dutton is a good minister, he's a hard worker, he's made a strong contribution to the government," she told the Nine Network on Monday.
"It's a solid sledge," she said, revealing she replied saying "you know mate, you've sent that mad witch text to the mad witch" and he immediately apologised. She added there were more substantive issues surrounding the Briggs affair - especially identifying the colleagues to whom he sent a photo of the public servant after their night in Hong Kong.
The public servant's details and photo were then leaked to the media and the image published in pixelated format in The Weekend Australian newspaper.
Mr Turnbull says the photo leak could deter victims of sexual harassment from coming forward.
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"Publishing the identity of a complainant in a case like this not only infringes their privacy, it serves actively to discourage other women who are concerned about the conduct of a superior from raising a complaint in the future," he told Fairfax Media.
Ms Maiden said Mr Dutton made the decision to own up to his text error after she chose not to name him when appearing on Sunday television about the column.
In a statement on Sunday, 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.
