The WikiLeaks founder entered the embassy in London in 2012 to avoid extradition to Sweden, where he was wanted in connection with sexual offences allegations.
He spent nearly seven years living in the embassy until being dramatically dragged out by police in April after Ecuador revoked his political asylum.
The 48-year-old was jailed for 50 weeks for breaching his bail conditions, just short of the one-year maximum for the offence.
He lodged an appeal against the length of his sentence and a hearing was due to take place at the Court of Appeal in London on July 23.
But Assange is no longer pursuing the appeal and a spokeswoman for the judiciary confirmed on Thursday that the planned hearing had been cancelled.
Sentencing him at Southwark Crown Court in May, Judge Deborah Taylor said it was "difficult to envisage a more serious example" of breaching the Bail Act.
Assange wrote to the court to apologise for his actions, which he said he regretted and acknowledged may have placed him in a more serious situation.
Judge Taylor said this was the first time he had expressed contrition over his actions, which she stated cost at least STG16 million ($A28 million) in public funds.
"Firstly, by entering the embassy, you deliberately put yourself out of reach, whilst remaining in the UK," she said.
"You remained there for nearly seven years, exploiting your privileged position to flout the law and advertise internationally your disdain for the law of this country."
She also said his actions "undoubtedly" affected Swedish prosecutors' efforts, which were discontinued in May 2017 "not least because you remained in the embassy".
In a handwritten letter to the court, Assange said he went into hiding while "struggling with terrifying circumstances".
"I apologise unreservedly to those who consider that I have disrespected them by the way I have pursued my case," he added.
"I did what I thought at the time was the best and perhaps the only thing that could be done, which I hoped might lead to a legal resolution being reached between Ecuador and Sweden that would protect me from the worst of my fears."
"I regret the course that this took.Whilst the difficulties I now face may have become even greater, nevertheless it is right for me to say this now."
Assange entered the embassy on June 19, 2012 while under intense scrutiny over the leaks of hundreds of thousands of classified US diplomatic cables on his whistle-blowing website.
The drastic move came after he exhausted all legal options in fighting extradition to Sweden over two separate allegations - one of rape and one of molestation.
Assange is fighting extradition to the US where he faces 18 charges including allegations of conspiring with intelligence analyst Chelsea Manning to hack into a classified Pentagon computer.
