Mr Tudge was also accused by his former staff member, Rachelle Miller, that he physically and emotionally abused her while in a consensual relationship. He has always denied the allegations.
Mr Tudge announced his resignation, which was to take effect at the end of next week, in the House of Representatives.
"My decision was cemented following the passing of my father," he said.
"It's not been an easy decision for me but it is necessary for my health and for my family, amongst other reasons."
Frydenberg rules out contesting Aston
His resignation will force a by-election, and prompted speculation that former treasurer Josh Frydenberg will seek Liberal Party endorsement for the seat.
But the ABC has confirmed Mr Frydenberg will not seek the seat.
Mr Tudge remained on the Coalition frontbench after last year's election, retaining the education portfolio.
His eastern Melbourne electorate was once a safe Liberal seat but turned marginal last year.
In 2019, the two-party preferred margin was 10.1 per cent, falling to 2.8 per cent in 2022.
Mr Frydenberg's previously represented the nearby, but not neighbouring, electorate of Kooyong, before losing to teal independent Monique Ryan.
He has since moved into the private sector.
Children faced death threats
Mr Tudge, a father of three, became emotional as he thanked his children for what they had endured while he was a cabinet minister.
He said his teenage children were subject to death threats.
Mr Tudge said he was honoured to have served in the federal parliament and believed the Liberal Party was well served in Aston to find a replacement.
"It truly has been an honour and privilege to be in this place," he said.
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That's it for Alan Tudge's resignation announcement, so we'll wrap up our live coverage here.
You can catch up on Thursday's developments as they happened below, or download the ABC News app and subscribe to our range of news alerts for the latest news.
ABC political editor Andrew Probyn says Alan Tudge's resignation places his electorate of Aston in a "precarious spot."
"Aston once was a fairly safe seat for the Liberals down in outer eastern Melbourne," he tells Afternoon Briefing.
"It is now just 2.6%. So a by-election at this particular point will not be particularly welcome for the Liberal Party.
"So there is a lot to be interested in when this by-election happens, probably about April."
"So I say to the member for Aston, we have political differences ... and there is no point gilding the lily there and he wouldn't want me to either, but I thank him for his service in this house," Anthony Albanese says.
"I thank him for the sacrifices that are made to serve in this house, and I, on behalf of the government, say that I respect the decision that he has made, and I respect the manner in which it has been made."
The Prime Minister said a few words following Alan Tudge's resignation.
"Serving in this house is an incredible honour for all of us, and it is a difficult day and a difficult decision for the member for Aston to come to," Anthony Albanese says.
He continues, dicussing how hard the role can be on the children of politicians, "who are sometimes exposed to what should never occur, particularly at a time where people can make anonymous comments through social media, and through other divisive forms."
"Mr Speaker, I leave honoured to have served, and proud of what we have achieved, both for the people of Aston and across Australia, but particularly for Indigenous Australia," Tudge says, wrapping up his speech.
"There is always much more work to do and there are always new faces in this place, but I believe that the Liberal Party is well served by the emerging talent in Aston, and it is time to pass the baton to someone who can hopefully serve our local community better than anyone before them.
"It truly has been an honour and a privilege to be in this place. I thank the house."
He walks back to his seat to applause.
He continues after taking a few sips of water, then joking by saying he's spilt it everywhere.
"I wanted to thank my family particularly, and particularly my kids. Cass, Chrissy and Richard, who I am so very proud of."
"I thank the Prime Minister and the Speaker for the grace they have shown me," Alan Tudge says.
"I especially thank Peter Dutton, Scott Morrison and Tony Abbott for their leadership, support and strength.
"Peter Dutton is a man of incredible integrity, experience and strength, and I greatly admire him, and he would make an excellent Prime Minister."
"While this profession can be unforgiving at times, serving as a member of parliament has been one of the greatest honours of my life," Alan Tudge continues.
"My passion has always been in social policy rather than economic security policy, the traditional Liberal focal areas.
"I've always believed that while the economy is the foundation of our society, that social policies determine whether individuals are given the opportunity and responsibility to realise their potential."
"My decision was cemented following the passing of my father," Alan Tudge says.
"It's not been an easy decision for me but it is necessary for my health and for my family, amongst other reasons."
"I certainly have not taken this decision lightly.
"My daughters, who are 18 and 16, no nothing other than their Dad being away.
"They've had to put up with things that no teenager should have to, including death threats.
"The most recent of which was last week."
Alan Tudge is speaking in Question Time in Canberra.
"I'd like to inform the house that I will be resigning from parliament effective from next week."
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