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.
One of the charges 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."
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." ....
Poor old Geoffrey Robertson has had to cut short his Australian holiday to head back to London!
