Earlier this month a High Court judge rejected Depp's claim that a newspaper committed libel when it called him a "wife-beater".

Judge Andrew Nicol said the article in The Sun was "substantially true".

Depp is seeking to overturn the judgment.

However in a setback for the Pirates of the Caribbean star, Nicol has denied permission to appeal, saying "I do not consider the proposed grounds of appeal have a reasonable prospect of success".

In a ruling made public on Wednesday, the judge also ordered Depp to make an initial payment of almost 630,000 pounds ($A1.1 million) to News Group Newspapers, publisher of The Sun, to cover legal fees.

Depp can still apply directly to the Court of Appeal before December 7.

The judge's main ruling came after a three-week trial in which Depp and Heard gave conflicting accounts of their brief, tempestuous marriage.

In the wake of the decision, Depp said he was leaving the Fantastic Beasts film franchise after studio Warner Bros. requested his resignation.

Depp is also suing Heard for $US50 million ($A68 million) in Virginia over a Washington Post op-ed essay she wrote about domestic violence.

The essay talks about her experience being abused but does not name Depp.

The trial is due to be held next year.

Australian Associated Press