Mr Assange denies sexually assaulting two women in Sweden. He was remanded in custody pending a hearing next week.
A judge at City of Westminster Magistrates' Court refused bail because of the risk of the 39-year-old fleeing.
A Wikileaks spokesman said the arrest was an attack on media freedom and pledged to continue publishing.
After the court appearance Mr Assange's lawyer Mark Stephens said he would be applying again for bail.
He claimed the accusations were "politically motivated" and said the judge was keen to see the evidence against Mr Assange, an Australian citizen.
Mr Stephens said Wikileaks would continue to publish material and added: "We are on cable 301 and there are 250,000 secret cables."
A Downing Street spokesman said Mr Assange's arrest was "a matter for the police" and there had been no ministerial involvement.
The Pentagon welcomed the arrest, with US Defence Secretary Robert Gates saying on a visit to Afghanistan that it was "good news".
Prosecutors in Sweden have insisted the extradition request is a matter of criminal law and they "have not been put under any kind of pressure, political or otherwise".
Five people, including journalist John Pilger, film director Ken Loach and Jemima Khan, the sister of Conservative MP Zac Goldsmith, offered to put up sureties.
But district judge Howard Riddle refused bail for Mr Assange and he was remanded in custody until 14 December.
Judge Riddle said he believed Mr Assange might flee and he also feared he "may be at risk from unstable persons".
Gemma Lindfield, for the Swedish authorities, gave details of the allegations against Mr Assange.
Profile: Julian Assange
One of the allegations is that he had unprotected sex with a woman, identified only as Miss A, when she insisted he use a condom.
Another is that he had unprotected sex with another woman, Miss W, while she was asleep.
Mr Assange, who was accompanied by Australian consular officials, initially refused to say where he lived but eventually gave an address in Australia.
Afterwards Ms Khan explained why she was willing to put up a surety: "I offered my support as I believe that this is about the universal right of freedom of information and our right to be told the truth."
By Clive ColemanBBC News legal affairs analyst
At a full hearing, which is not likely to take place for some weeks, Mr Assange will be able to raise his arguments against extradition.
The "fast-track" European arrest warrant system is based on the concept that all the participating countries have legal systems which meet similar standards, and fully respect human rights.
If Julian Assange is to avoid extradition he would need to show the warrant is politically motivated. This has been argued successfully in the past by Russian oligarchs, though Sweden has a better judicial record than Russia.
Or he would need to use technical arguments - arguing the warrant does not show specifically what law has been broken. But most technical mistakes could be resolved eventually and the warrant reissued.
Wikileaks spokesman Kristinn Hrafnsson said it would not stop the release of more secret files and told Reuters on Tuesday: "Wikileaks is operational. We are continuing on the same track as laid out before.
"Any development with regards to Julian Assange will not change the plans we have with regards to the releases today and in the coming days."
He said Wikileaks was being operated by a group in London and other secret locations.
Secret locations
US State Department spokesman PJ Crowley said it was possible the US would make an extradition request for Mr Assange but he said it was premature as the criminal investigation into Wikileaks was still ongoing.
Mr Assange was arrested by appointment at a London police station at 0930 GMT.
Wikileaks' struggle to stay online
Police contacted Mr Stephens on Monday night after receiving a European arrest warrant from the Swedish authorities.
An earlier warrant, issued last month, had not been filled in correctly.
Mr Assange has come in for criticism in the past week for the revelations made on Wikileaks.
Foreign Secretary William Hague has criticised the website for publishing details of sensitive sites, including some in the UK, saying they could be targeted by terrorists.
But, in an article for The Australian newspaper, Mr Assange denied he had put lives at risk and said Wikileaks deserved to be protected.
Mr Crowley said: "The (US) constitution enshrines the freedom of the press and we respect that, even if we have concerns about how that is exercised".
Gerard Batten, a UKIP MEP, said the Assange case highlighted the dangers of the European arrest warrant.
He said: "I don't know of the quality of the evidence in Mr Assange's case but it does seem that he is involved in political turmoil and intrigue and there are a lot of people keen to shut him up and there is nothing a court in the UK can do to look at the evidence before they extradite him."
Mr Assange's supporters have written an open letter to Australian Prime Minister Julia Gillard, asking her to protect him.
Mr Pilger, who appeared in court to support Mr Assange, said Ms Gillard's threat to remove his passport smacked of "totalitarianism".
The founder of whistle-blowing website Wikileaks, Julian Assange, has been refused bail by a court in London but vowed to fight extradition to Sweden.