Important lessons must be learned from an independent probe into claims of child sex abuse by Sir Jimmy Savile, Labour's deputy leader has said.
Harriet Harman warned it would be "complacent" to assume "systematic" abuse "would never happen now".
Her comments back leader Ed Miliband's call for a wider inquiry covering the BBC, NHS and Broadmoor secure hospital.
Police say ex-presenter Savile may have sexually abused 60 people since 1959. The BBC has ordered two reviews.
No 10 said the police investigation into allegations against Savile, who died in October 2011, was a priority. However, it did not rule out a wider inquiry in the future.
A raft of allegations regarding the former BBC DJ and presenter's conduct came to light in the wake of an ITV investigation, which was broadcast on 3 October.
Scotland Yard, which is co-ordinating the investigation into Savile's alleged offences, has said it is following up 340 lines of inquiry.
The BBC has been criticised for not calling Savile's behaviour into question or flagging up any abuse allegations during his long career at the corporation, during which he presented several television shows including Top of the Pops and Jim'll Fix It.
Abuse is also alleged to have taken place at high-security psychiatric hospital Broadmoor and at Stoke Mandeville hospital and Leeds General Infirmary, where Savile volunteered.
The Department of Health has said it will investigate its own conduct in appointing Savile to lead a taskforce overseeing the management of Broadmoor in 1998.
Leeds Teaching Hospitals Trust has said it would help police if asked to do so.
The BBC has launched two separate internal investigations into the Savile affair. The first will look at why a BBC Newsnight investigation into abuse claims was not brought to air.
The second will look into the culture and practices of the BBC during the years that Jimmy Savile worked at the corporation, and afterwards. This will only start once police have indicated they are happy for it to proceed.
'Dark side of celebrity'
Speaking to the BBC, Ms Harman underlined that "serious abuse" carried out by Savile had involved "a spate of organisations - first and foremost obviously the BBC, but not just the BBC".
"Instead of having a number of separate, different investigations, there should be one inquiry across the piste because there is a systematic problem here.
"But above all, as Ed Miliband said, it is important for the victims to know there is an independent inquiry - not that each organisation is looking into itself."
She added: "I think the remit (of an independent inquiry) would be to learn the lessons and to make sure that it couldn't happen again, because I don't think we can be complacent [and say], 'Oh that was 20 years ago so actually it would never happen now.'
"I think we need to be certain about that and that's why I think we need an inquiry."
Earlier, Mr Miliband told ITV's The Agenda that internal investigations were not enough. He said: "In order to do right by the victims, I don't think the BBC can lead their own inquiry.
"We need a broad look at all the public institutions involved - the BBC, parts of the NHS and Broadmoor. This has got to be independent."
The Labour leader added the abuse was "absolutely horrific" and would "scar [the victims] for life".
Meanwhile, a West Yorkshire MP has asked the TV watchdog to investigate whether the BBC should be allowed to hold a broadcasting licence.
Shipley's Philip Davies has written to the head of Ofcom saying it has become "increasingly obvious" staff working for the BBC knew about allegations of child sex abuse by Savile.
Speaking in the Commons, Deputy Prime Minister Nick Clegg said there "may be a case" for an independent inquiry into the abuse claims at the BBC and other organisations.
However, he said an inquiry should not be set up until it was clear it would not be restricted by the police investigation into the claims of sexual abuse.
Mr Clegg also said the abuse was "in many ways the dark side of celebrity".
BBC director general George Entwistle has offered to appear before the House of Commons culture, media and sport select committee over the scandal.
He had been due to appear in front of MPs later this year but had offered to bring it forward to 23 October, said culture committee chairman John Whittingdale.