"We conclude that the appeal has no real prospect of success and that there is no other compelling reason for it to be heard."
The Hollywood actor launched legal action against News Group Newspapers over an article published in April 2018.
Amber Heard Alleges Johnny Depp ‘Threatened To Kill Her Many Times’, As She Begins Evidence In Libel Case
Depp is suing The Sun's publisher over an article from 2018.
The actor's legal team said there is "no need" to hear from his former partners who were both due to give evidence.
The actor is facing a fourth day of questioning during his high-profile libel case against The Sun.
The actor denied assaulting his ex-wife during a trip to Australia, as he faced his third day of questioning at his libel trial against The Sun.
The actor is currently suing The Sun's publisher over an article published in 2018 describing him as a "wife beater".
The actress' lawyer calls the allegation "absurd, offensive and categorically untrue".
The former husband and wife have accused each other of being abusive.
"We will prevail in defeating this groundless lawsuit..."
Heard has discussed the aftermath of the split.
ason Momoa, Amber Heard and other stars took to the blue carpet in London for the premiere of ‘Aquaman’. Aquaman is one of the characters in DC Comics' "cinematic universe", having also appeared in "Batman v Superman: Dawn of Justice" and "Justice League". The character has long been seen as a joke in the superhero world. This will be his first solo outing.
"Aquaman" actress Amber Heard criticises a Daily Mail article which called her recent outfit choice a "wardrobe malfunction".
"Stop dramatising his behaviour as though he is some sort of misunderstood hero."
“Amazing,” said the Seattle shop owner who sold the headpiece to Heard’s stylist.
The actress’ name was included on a list of sizable donations to the Children’s Hospital Los Angeles.
Allegations of domestic violence were first made by Amber in May 2016, when she sought a temporary restraining order against
“I can say that my private life is my private life. Or I could not let go and own it.”
The tables have turned on Barnaby Joyce.