Mr Tudge's former media advisor Rachelle Miller alleged he had been emotionally and in one instance, physically abusive towards her during the pair's consensual affair.
Mr Tudge denied any wrongdoing, but told the prime minister he would not be seeking to return to the front bench "in the interests of his family and his own wellbeing and in order to focus on his re-election as the member for Aston".
Prime Minister Scott Morrison said Dr Vivienne Thom, who investigated the allegations, found that the evidence in the inquiry "does not provide a basis for a finding that Mr Tudge's conduct breached the ministerial standards".
Ms Miller did not participate in the inquiry, saying the process was flawed and the government had ignored her concerns about the terms of reference.
Ms Miller said the minister repeatedly chastised, bullied and belittled her, outlining methods of coercive control.
On one occasion, she alleges Mr Tudge kicked her until she left his bed after her phone rang early one morning and he became frustrated it woke him.
Both parties have been supplied with a copy of the report.
Stuart Robert will continue in his role as acting education minister.
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