Peter Watt, the NSPCC's director of national services, said:
According to the report, Would They Actually Have Believed Me?, some of the victims, who were aged between eight and 26 when Savile assaulted them, told hospital staff, who dismissed their claims.
One of the 26 victims interviewed by NSPCC counsellors went to the police but no action was taken. The vast majority were children when they were abused but four were adults.
The NSPCC said the research, which was commissioned by Her Majesty's Inspectorate of Constabulary, highlighted the "devastating scars" that victims had suffered from the abuse, with some turning to drink and drugs to cope.
Others have suffered mental illness, poor relationships or contemplated suicide, it said.
Head of Child Protection Operations at the NSPCC, John Cameron, says children have a "greater likelihood" to be involved in criminal and anti-social behaviour if "they have been in receipt of smacking as a standard form of parenting".
Speaking with ITV News, Mr Cameron said smacking does not have a positive, long-term impact on children.
"This is not about prosecuting, persecuting and criminalising parents, but it's about giving a very strong message in our society that children should have the same rights as adults in law to be protected from physical assaults," he said.
The NSPCC has welcomed Google and Microsoft's decision to block images of child abuse via their search engines, but emphasised, "We must keep one step ahead."
Peter Wanless, chief executive of the children's charity, said:
The NSPCC is calling for social media sites to do more to make their online platforms safer for the millions of children who use them by bypassing the age restrictions. Claire Lilley, NSPCC's head of child safety, said the risks to 11 and 12-year-olds need to be evaluated by the companies.
BeatBullying has backed calls from the NSPCC for social media sites to do more to protect their young users.
BeatBullying founder Emma-Jane Cross said:
"This research underlines the urgent need for social networking sites to take their young users' safety much more seriously.
"We believe in the right of children to go online without fear of being bullied or harassed, and this cannot be achieved until social networks put adequate resources behind safeguarding and user verification policies."
The NSPCC are calling for social media sites to acknowledge that many of their users are under the age of 18, in order to do more to safeguard their child users.
The comments come as new research by the children's charity found that around half of the UK's 11 and 12-year-old regularly use social media, and of these, a quarter said they had been upset by something on it.
A quarter of children aged between 11 and 12 who use social media have been upset by something on it over the past year, new research from the NSPCC has revealed.
A poll of more than 1,000 youngsters also found that a fifth of children who had been upset by an online incident such as trolling, bullying or being sent inappropriate sexual messages, experienced this every day or almost every day.
Justice Minister Damian Green admitted more work was needed to help children be supported through the court process. Responding to the NSPCC report that highlighted failures in adequate care of young and vulnerable witnesses at criminal proceedings, he said:
A senior policy analyst for the NSPCC says young witnesses can be questioned by barristers in an "aggressive way" when giving evidence in court in sexual abuse cases.
Lisa McCrindle told the BBC: