Newsnight reporters had began an inquiry into allegations of inappropriate behaviour by Savile towards teenage girls taking place in his 1970s heyday.
Friends and family of the former Jim’ll Fix It star accused the BBC’s journalists of “muckraking”.
Surrey police had previously investigated the claims, which were made in 2007, and Crown Prosecution Service lawyers concluded that there was no evidence to press charges.
The former DJ, broadcaster and charity fundraiser died at his home in Leeds in October last year.
The eccentric character never married and his private life was the subject of gossip and speculation.
The BBC2 programme’s investigation focused on three women who met Savile in the 1970s when he presented Top of the Pops.
Teenagers at the time, they attended a school in Surrey that Savile visited as part of his charity work.
The three women all say that he took them out individually in his car and behaved inappropriately.
But senior BBC executives decided to scrap the show, which was due to take place a few days before a Christmas special celebrating Savile’s work.
A spokesman for the Crown Prosecution Service said: “Following an investigation by Surrey Police, the CPS reviewing lawyer advised the police that no further action should be taken due to lack of evidence.”
A BBC spokesman said: “Any suggestion that a story was dropped for anything other than editorial reasons is completely untrue.
The BBC gathers information on hundreds of stories and not all make it to air. In this case the angle we were pursuing could not be substantiated.”
Stephen Purdew, a friend of the star who helped organise his funeral, said: “These are allegations from 40 years ago. He’s not even alive to defend himself.
“Sir Jim was a great man, a legend who should be remembered for all the wonderful things he did for other people.”
The cigar-loving star whose catchphrase was “now then, now then”, gave away 90 per cent of his income to good causes.
Thanks to this, he was knighted and formed a close relationship with members of the Royal Family and politicians including former prime minister Margaret Thatcher.
BBC staff are said to be angry that the show was pulled, having spent months working on the report.
A source said: “Someone, somewhere should have realised that a report trying to uncover a dark side of one of the BBC’s favourite sons was a silly thing to do.”