Following a three-week trial in July last year, Mr Justice Nicol dismissed the Hollywood star’s libel claim against the publisher of The Sun, finding that an April 2018 column calling Mr Depp a “wife beater” was “substantially true”.
The judge ruled Mr Depp, 57, assaulted Ms Heard, 34, on a dozen occasions and put her in “fear for her life” three times.
The actor asked the Court of Appeal to grant permission for him to challenge the ruling, with the aim of having its findings overturned and a retrial ordered.
But on Thursday, the court refused permission for Mr Depp to appeal the High Court ruling.
Announcing the ruling at the Royal Courts of Justice in London, Lord Justice Underhill said: "We refuse Mr Depp’s application to admit further evidence in support of his proposed appeal and we conclude that the appeal has no real prospect of success and that there is no other compelling reason for it to be heard.
"We accordingly refuse permission to appeal."
Lord Justice Underhill, sitting with Lord Justice Dingemans, added: "As we have said, it is not easy to persuade this court to overturn the findings of a trial judge on purely factual questions.
"We do not believe that there is a real prospect of it being prepared to do so in this case.
"The hearing before (Mr Justice) Nicol was full and fair, and he gave thorough reasons for his conclusions which have not been shown even arguably to be vitiated by any error of approach or mistake of law."
In a statement after the ruling, a spokeswoman for The Sun said: "The Sun had every confidence that this leave to appeal application would not be granted and are pleased with today’s decision.
"The case had a full, fair and proper hearing, and today’s decision vindicates the courageous evidence that Amber Heard gave to the court about domestic abuse, despite repeated attempts to undermine and silence her by the perpetrator.
"The Sun will continue to stand up and campaign for victims of domestic abuse."
At a hearing last week, Mr Depp’s lawyers asked the court to consider fresh evidence relating to what they said was Ms Heard’s claim that she gave her $7 million (£5.5 million) divorce settlement to charity.
His barrister Andrew Caldecott QC told the court that claim was a "calculated and manipulative lie".
After the couple divorced in 2016, Ms Heard said she would split the settlement between the Children’s Hospital Los Angeles and the American Civil Liberties Union (ACLU).
But, Mr Caldecott said, the hospital wrote to Mr Depp’s business adviser in 2019 to say Ms Heard had not made "any payments".
The court heard $100,000 (£72,000) was donated to the hospital and $450,000 (£322,000) to the ACLU, although Ms Heard claims she made a further $500,000 (£358,000) donation to the second charity anonymously.
Mr Caldecott said the claims had given Ms Heard "a considerable boost to her credibility as a person", and had "tipped the scales against Mr Depp from the very beginning".
But Adam Wolanski QC, representing The Sun’s publisher News Group Newspapers (NGN), said the new evidence Mr Depp wanted to rely on "would not have had any impact" on the result of the trial.
He also rejected Mr Depp’s contention that Ms Heard had lied about donating her divorce settlement to charity, saying: "The information does not demonstrate that Ms Heard lied."
Mr Depp sued NGN in June 2018 over the column by The Sun’s executive editor Dan Wootton, which referred to "overwhelming evidence" he attacked Ms Heard.
In his judgment, Mr Justice Nicol concluded 12 of the 14 alleged incidents of domestic violence relied on by NGN in its defence of the actor’s claim did occur.
The judge also found Mr Depp put Ms Heard in "fear for her life" on three occasions, including one the actress described as a “three-day hostage situation” in Australia in March 2015.
Days after the ruling in November, Mr Depp announced he had been asked by Warner Brothers to resign from his role in the Harry Potter spin-off franchise Fantastic Beasts.
Mr Depp is embroiled in a separate libel battle in the US, having sued Ms Heard personally over a 2018 Washington Post opinion piece in which she claimed to be a victim of domestic abuse but did not mention the actor by name.
The actor’s $50 million (£35 million) US case against Ms Heard has been delayed until April 2022.