In December 2010, Assange was accused of raping one woman and molesting and coercing another. Two Swedish women made accusations after his visit to Stockholm in August.
In preparation for an appeal, Assange's legal team needs to seek permission from the High Court. They will have 14 days to do so and bring the case to the Supreme Court, on the grounds his case raises issues of public importance.
Assange sat silently through the judgment and gave no sign of emotion as the judges gave reasons for their decision, according to reports.
"It is clear that the allegation is that he had sexual intercourse with her when she was not in a position to consent and so he could not have had any reasonable belief that she did," the judges said.
Although the President of the Queen's Bench Division Sir John Thomas said the issuing of the European arrest warrant (EAW) was "lawful" and "proportionate", 40-year-old Assange denies allegations, claiming they are politically motivated.
"I have not been charged with any crime in any country," said Assange in his speech outside the High Court after the hearing.
He also urged people to go to swedenversusassange.com to "know what is really going on in this case."
Supporters outside the court held banners reading "Free Assange! End the wars" and vented their outrage at the judges' decision.
Last month, WikiLeaks pleaded for funding so it could continue to publish Government diplomatic cables after major financial institutions cut off support for the organisation.