Porter said he had waited until the NSW police concluded the case so that he did not interfere with the investigation.
"Staying silent. Following the rules. A very difficult decision. I have been subject to the most wild, unrestrained allegations in Australian politics."
The Attorney-General said that he was "deeply sorry" that the speculation had involved his colleagues who were "his friends".
Porter said he had been subjected to trial by the media when he was bound to remain silent.
"I have given the bulk of my adult working life to public service and the law. I have given absolutely everything I had in the tank over the last year to our government, which has been desperately trying to help the country out of the worst crisis in its modern history,'' he said.
"If I stand down from my position as Attorney-General because of an allegation about something that simply did not happen, then any person in Australia can lose their career, their job, their life's work based on nothing more than an accusation that appears in print.
"If that happens, anyone in public life is able to be removed simply by the printing of an allegation. Every child we raise can have their lives destroyed by online reporting of accusations alone.
"My guess is if I were to resign and that set a new standard there wouldn't be much need for an Attorney-General anyway because there would be no rule of law left to protect in this country, so I will not be part of letting that happen while I am Attorney-General."
Porter said he would now take a period of mental health leave.
"After speaking with my own doctor I am going to take a short period of leave to assess and hopefully improve my own mental health,'' he said.
"All of my life I have just pushed through, but for the many caring family and friends who have asked me that question over the course of the last week, "Are you OK?" I have got to say my ... answer is I really don't know. I am not ashamed to say that I am going to seek some professional assessment and assistance on answering that question over the next few weeks."
Porter, 49, made the statement after consulting defamation lawyers.
The woman was an award-winning debater who met him when she represented South Australia on a state debating team as a teenager. In an unsworn statement she claimed the incident happened in 1988 after a night of drinking and dancing in Kings Cross.
At the time, she was 16 years old. Porter was 17.
The claims were circulated last week in an anonymous dossier circulated to police and political leaders including Prime Minister Scott Morrison, former Prime Minister Malcolm Turnbull and Greens Senator Sarah Hanson-Young.
The woman who made the claim died in June 2020 after taking her own life in Adelaide, having approached police in 2019. Her family are believed to remain adamant they do not want her identified.
On Tuesday, NSW police confirmed they will not proceed with an investigation.
"NSW Police have since sought legal advice in relation to these matters,'' a spokesman said.
"Based on information provided to NSW Police, there is insufficient admissible evidence to proceed.
"As such, NSW Police Force has determined the matter is now closed."
The Prime Minister has previously revealed the Liberal minister at the centre of the allegation, who at that stage had not been identified, "vigorously" denies the allegations.
But former Prime Minister Malcolm Turnbull said on Tuesday that the minister must now "out himself" in respect for his colleagues and the country.
"He should out himself and he should provide a comprehensive statement," he said.
"He should describe when he knew the woman, how he knew the woman, what dealings he had with the woman after the event. We need to know what he knew about the complaint and when he knew about it.
"Frankly, it's not good enough for the Prime Minister to say 'Oh, it's a matter for police'. The Prime Minister cannot outsource his responsibility for composing his ministry to the police.
"He should require the minister to speak up. He owes it to his colleagues and the country."
The NSW police investigation into the Adelaide woman's claims was paused by her just days before her suicide.
However, SA Police are going through evidentiary material relating to her death to ensure all the documentation is sent to a coronial inquiry.
On Monday, Morrison said he first heard about an anonymous letter to him detailing the claims last week and spoke to the minister and the AFP commissioner that same night.
The Prime Minister also revealed he had been "briefed" on the unsubstantiated rape allegations against the minister but that he had not read the document or the woman's own words.
"Did I raise it? Yes, I did. And he vigorously and completely denied the allegations. So that means there is a proper process now for it to follow,'' Morrison said.
"It is the police, in a country where you're governed by the rule of law, that determine the veracity of any allegations of this nature,'' he said.
Asked if he believed the minister's denials Morrison said that was a matter for the police.
Lawyer Michael Bradley, who acted for the woman before her death, has called for an independent political inquiry into the matter.
"It's a very unusual and unfortunate situation," Bradley said.
