David Cameron, Ed Miliband and Nick Clegg have called for the Press Complaints Commission to be scrapped following its handling of the News of the World (NoW) phone hacking scandal.
The prime minister called the watchdog "ineffective and lacking in rigour" and demanded "a new system entirely".
The Labour leader said the organisation was a "toothless poodle".
But public members of the PCC said the government could not simply order its replacement as it was independent.
Former NoW editor Andy Coulson has been arrested by police investigating corruption and phone hacking claims.
Mr Coulson worked as Mr Cameron's communications chief after resigning from the NoW in 2007, following the jailing of the paper's royal editor and a private investigator over phone hacking.
Mr Miliband said the prime minister had "serious questions" to answer about this decision, which he called an "appalling error of judgement".
Tony Blair, who was prime minister at the time of the alleged phone hacking, described the scandal as "beyond disgusting" and called for a major public debate about the media's role in society.
Mr Cameron said he took "full responsibility" for hiring Mr Coulson, who quit his government job earlier this year, saying he had decided to "give him a second chance".
"People will be able to judge whether that was the right or the wrong thing to do," he said.
Speaking in Downing Street, he added that he had sought assurances from Mr Coulson about his past activities before recruiting him but the "second chance had not worked out".
Asked about the future of former NoW editor Rebekah Brooks, who is now chief executive of the paper's parent company News International, Mr Cameron said it was not his job to "pick and choose" who ran newspapers.
But he said he understood that she had offered her resignation and "I would have accepted it".
Mr Cameron said any new press regulatory body should be "truly independent", unlike the PCC, which is part-run by newspaper editors.
In a 2009 report the PCC concluded there was no evidence it had been misled over phone hacking by the NoW, which is closing down this weekend.
The prime minister said: "Let's be honest. The Press Complaints Commission has failed. In this case, the hacking case, frankly it was pretty much absent.
"Therefore we have to conclude that it's ineffective and lacking in rigour.
"There is a strong case for saying it's institutionally conflicted because competing newspapers judge each other. As a result it lacks public confidence.
"I believe we need a new system entirely. It will be for the inquiry to recommend what the system should look like."
In a speech in London, Mr Miliband said the impending closure of the NoW, following allegations that crime victims and bereaved families may have had their mobile phones hacked into, was "not the answer" to restoring trust and senior executives must "take responsibility".
He added: "For too long, the political class have been too concerned about what people in the press would think and too fearful of speaking out. We must all bear responsibility for that, my party has not been immune from it, nor has the current government."
While defending the right to a "free and buccaneering press", he said newspapers "must reform in order to protect and restore their reputation".
Mr Miliband has called for Mrs Brooks - who was editor of the NoW at the time it is alleged murdered schoolgirl Milly Dowler's phone was hacked into - to resign as chief executive of News International.
But he said in his speech that "wholesale reform of our system of regulation" was also now needed.
He added: "The Press Complaints Commission has totally failed. It failed to get to the bottom of the allegations about what happened at News International in 2009.
"Its chair admits she was lied to but could do nothing about it. It was established to be a watchdog. But it has been exposed as a toothless poodle. It is time to put it out of its misery.
"The PCC has not worked. We need a new watchdog. There needs to be fundamental change."
Former Prime Minister Mr Blair said: "Anyone who has been a political leader in the last four decades knows really that there is this huge debate that should take place about the interaction between the media and politics and the media and public life."
Speaking at an event in Westminster, he said he was "vilified" for describing the press as "feral beasts" in a speech four years ago, but said the debate could now happen.
He added: "Both Ed Miliband and David Cameron are right to say this is not just about News International. It is not just about phone hacking."
Deputy Prime Minister Nick Clegg said the PCC was a "busted flush" and needed to be replaced as part of a review of the "whole architecture of the way the press operates". No newspaper could act "with complete impunity, as if there is one law for them and one law for everyone else", the Lib Dem leader added.
But independent, public members of the PCC said those calling for its demise had undervalued and underestimated its day-to-day work.
In a statement they said: "The government cannot simply order the replacement of the PCC, because it is an independent organisation.
"While we publicly accept the need for reform and improvement, and welcome the inquiries, our highly important work must still go on."
They said the "vital" work of staff must be "built upon, not jettisoned".
Julie Spence, a former chief constable and now a lead on the PCC panel investigating phone hacking, insisted the commission must not become "the sole political scapegoat for systemic failings across the board".
"The commission will take its share of responsibility but that is not the lion's share."
She defended the current Lay commissioners, among whom are a solicitor, a retired judge and an ex-BBC chairman.
"All are independent thinkers, none are shrinking violets and all want and are capable of being part of the solution.
"We are also advocates of a free but responsible press and are committed to achieving this," she added.
Bob Satchwell, executive director of the Society of Editors, said the idea that the NoW scandal had shown up a failure of ethics across the industry was "total nonsense".
The PCC, set up in 1991, includes the editors of the Mail on Sunday, Sunday Telegraph and Sunday Mirror among its 17 members.