The 77-year-old singer sued over BBC coverage of a South Yorkshire Police search of his home in Sunningdale, Berkshire, in August 2014, following a child sex assault allegation.
Earlier this month Mr Justice Mann ruled in Sir Cliff's favour following a High Court trial in London.
The judge concluded the BBC's coverage was a "very serious" invasion of the singer's privacy and awarded Sir Cliff £210,000 damages.
He said the award would be made up of £190,000 to cover the "general effect" coverage had on Sir Cliff's life, plus £20,000 because the BBC had aggravated harm by nominating coverage for an award.
The BBC was refused permission to challenge the ruling by Mr Justice Mann at a hearing in July.
In a statement on Wednesday, the corporation announced it would no longer be pursuing an appeal against the judgment.
Instead, Director General Tony Hall has written to the Attorney General Geoffrey Cox QC and asked him to consider a review of the law on naming people involved in police investigations.
The organisation reiterated its view that the ruling "represents a dramatic shift against press freedom".
Mr Justice Mann previously said his judgment did not amount to a "blanket" restriction on journalists.
The BBC repeated its apology to Sir Cliff for the distress caused and said: "We fully appreciate the impact this has had on him.
"There are lessons for the BBC in how we reported this story and we will think very carefully about our approach in the future - both in tone and style.
"We recognise there are things we got wrong - even if all the facts we reported were right."
A spokesman for Sir Cliff said: "Sir Cliff reluctantly took his case to court because he felt his privacy had been flagrantly invaded and disappointingly the BBC were not prepared to acknowledge that and apologise.
"He welcomes the fact the BBC have decided not to seek permission to appeal from the Court of Appeal, particularly after the judge gave his judgement that they had no grounds on which to pursue such an action.
"Sir Cliff now hopes that outstanding issues can be resolved quickly."
Stay up to date with the latest news stories from a Christian perspective. Sign up to our daily newsletter and receive more stories like this straight to your inbox every morning.
