The Foreign Ministry made the request so that Assange, who has been living inside the London embassy for over three years, could undergo an MRI scan.
Assange faces arrest if he leaves the building although police have now removed a 24 hour guard.
In a press conference, Ecuadorian foreign minister Ricardo Patino read out a letter from Assange’s UK doctor who conducted a medical examination in August.
The letter says he is in constant and severe pain, which is growing worse and has been present since June this year.
The doctor said that an MRI scan needs to be performed which can only be carried out in a hospital.
The letter states: “He has been suffering with a constant pain to the right shoulder region for the past three months [since June 2015]. There is no history of acute injury to the area. I examined him and all movements of his shoulder (abduction, internal rotation and external rotation) are limited due to pain.
“I am unable to elicit the exact cause of his symptoms without the benefit of further diagnostic tests, [including] MRI.”
The source of the medical condition can only be diagnosed with hospital equipment that cannot be brought into the embassy due to its size and weight.
Ecuador wrote to the UK’s Foreign and Commonwealth Office on September 30 to request that Assange be permitted to go to hospital.
Ecuador said the safe passage it was seeking would be for a few hours to allow Assange to have medical tests.
The Ecuador government said the Foreign Office had rejected the request. Assange has been granted political asylum by Ecuador.
He is seeking to avoid being extradited to Sweden where he faces a sex allegation, which he denies.
He fears that if he goes to Sweden he will be taken to the United States for questioning over the activities of WikiLeaks.
His US lawyer Carey Shenkman said: “By claiming that Mr Assange must give up his asylum in order to receive medical treatment, the UK government is forcing him to choose between the human right to asylum and the human right to medical treatment.
“No one should ever have to face that choice. Sweden and the United Kingdom have the responsibility to ensure that Assange’s basic rights are respected. They should agree without further delay to permit Mr Assange’s safe passage to a hospital on humanitarian grounds.”