He denied the allegation, labelling it "spurious and defamatory", The Guardian reports, but acting Prime Minister Matthias Cormann described it as "extremely serious". "This has got to stop", he told reporters yesterday in Armidale, a New South Wales town in the New England electorate Joyce represents.
"I have asked for the right of that person who's made the allegation and I've asked for my right to defence that that be referred to police", he said.
"It's quite evident that you can't go to the despatch box with issues like that surrounding you". The prime minister spoke of the "terrible hurt and humiliation that Barnaby, by his conduct, has visited on his wife Natalie and their daughters - and indeed his new partner". Mr Joyce described his leader's remarks as "inept" and "completely unnecessary".
Mr Joyce, a political maverick who has led his party since elections in mid-2016, said he would not accept any ministerial portfolio.
Some Nationals have told the ABC privately they are frustrated by the damage Mr Joyce has already done to his party.
In response to the scandal, the prime minister announced a ban on government ministers having sex with their employees and later called Joyce's actions a "shocking error of judgement".
Mr Joyce explained his decision to quit the frontbench, saying there had to be some "clear air".
Mr Joyce still enjoys strong support among some of his colleagues who believe he is their best electoral asset.
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On Friday, Joyce finally said he would resign, after a separate and formal sexual harassment complaint made against him to the National Party emerged.
In a statement from Washington, Turnbull Friday thanked Joyce for being "a fierce advocate for rural and regional Australia", while insisting the Nationals-Liberal coalition was "undiminished" by the scandal.
But Mr Turnbull was effectively unable to sack Mr Joyce, whose position as party leader and Deputy Prime Minister is decided by National MPs.
"I think it's my responsibility to do my bit to make sure that happens", Joyce said.
Former Cabinet Minister Darren Chester announced he too would back Mr McCormack and ruled out making a bid himself.
"The party will greet this news with a heavy heart but we understand and respect his decision to stand down as leader", Mr Anthony said.
Nick Economou, senior lecturer in Australian politics at Monash University in Melbourne, said it would have been impossible for both Turnbull and Joyce to return to parliament next week after they fell out.