The 39-year-old Australian is wanted by prosecutors in Sweden over claims he sexually assaulted two women.
Jemima Khan, the sister of Tory MP Zac Goldsmith, film director Ken Loach and veteran journalist John Pilger all offered to stand as surety for Assange.
Gemma Lindfield, for the Swedish authorities, told the court Assange was wanted in connection with four allegations.
She said the first complainant, Miss A, said she was victim of "unlawful coercion" on the night of August 14 in Stockholm.
The court heard Assange is accused of using his body weight to hold her down in a sexual manner.
The second charge alleged Assange "sexually molested" Miss A by having sex with her without a condom when it was her "express wish" one should be used.
The third charge claimed Assange "deliberately molested" Miss A on August 18 "in a way designed to violate her sexual integrity".
The fourth charge accused Assange of having sex with a second woman, Miss W, on August 17 without a condom while she was asleep at her Stockholm home.
Assange, who appeared pale but calm in the dock, confirmed his name and date of birth at the request of the court clerk.
There was confusion over his address as he asked whether it was "for correspondence or for some other reason".
Assange, who was accompanied by officials from the Australian High Commission, eventually gave an address in his native country.
The one-hour court hearing came just hours after Assange was arrested by appointment when he attended a central London police station.
A European Arrest Warrant issued by the Swedish authorities was received by officers at the Metropolitan Police extradition unit last night.
An earlier warrant, issued last month, was not valid as officials had failed to fill in the form properly.
The court heard that the "household names" were prepared to stake their reputation by offering sureties with a total value of £180,000.
Mr Loach, who offered £20,000, said he did not know Assange other than by reputation, but he said: "I think the work he has done has been a public service.
"I think we are entitled to know the dealings of those that govern us."
Mr Pilger, who also offered £20,000, said he knew Assange as a journalist and personal friend and had a "very high regard for him".
He said: "I am aware of the offences and I am also aware of quite a lot of the detail around the offences.
"I am here today because the charges against him in Sweden are absurd and were judged as absurd by the chief prosecutor there when she threw the whole thing out until a senior political figure intervened."
Ms Khan offered a further £20,000 "or more if need be" although she said she did not know Assange.
Assange leaned forward and listened intently as the case against him was outlined to the court.
Mrs Lindfield said he should not be granted bail because there was a risk he would fail to surrender and also for his own protection.
She outlined five reasons why there was a risk: his "nomadic" lifestyle, reports that he intended to seek asylum in Switzerland, access to money from donors, his network of international contacts and his Australian nationality.
Mrs Lindfield added: "Any number of people could take it upon themselves to cause him harm.
"This is someone for whom, simply put, there is no condition, even the most stringent that would ensure he would surrender to the jurisdiction of this court."
Assange's solicitor John Jones said the case was not about WikiLeaks but was a "simple accusation" case with the right to bail.
He said his client was of good character and had voluntarily surrendered to police for his arrest.
Mr Jones said: "In relation to the state of play in Sweden, it is important for the court to be aware of the background to this. Mr Assange has made repeated requests that the allegations against him be communicated to him in a language he understands.
"That has been ignored by the Swedish prosecutor. Another Swedish prosecutor dropped this case early on for lack of evidence and it was resurrected in Gothenburg rather than Stockholm."
Mr Jones said his client resisted extradition because it is "disproportionate" to extradite someone under such circumstances.
He denied Assange is living a nomadic lifestyle and told the court he failed to provide a DNA sample and fingerprints on legal advice.
He added that it was not that Assange was "uncooperative" and he denied that his client was planning to seek asylum in Switzerland.
The court heard Assange did not have access to vast sums of money and had experienced significant difficulties with his finances.
District Judge Riddle said: "This case is not, on the face of it, about WikiLeaks. It is an allegation in another European country of serious sexual offences alleged to have occurred on three separate occasions and involving two separate victims.
"These are extremely serious allegations. From that, it seems to me that if these allegations are true, then no one could argue the defendant should be granted bail.
"If they are false, he suffers a great injustice if he is remanded in custody. At this stage in these proceedings, the nature and strength of the allegations is not known."
Remanding him in custody, the district judge said Assange faces "serious" allegations and has "comparatively weak community ties in this country".
Downing Street said that Mr Assange's arrest was "a matter for the police".
Asked what ministerial involvement there had been in the extradition request, a spokeswoman replied: "None."
The court heard that Assange spent two months staying at the Frontline Club, a media club in Paddington, central London, and is now living with a woman.
His solicitor said he would be "instantly recognised" if he tried to leave the country and would be "instantly apprehended".
He also argued that there was as much risk that "unstable persons" might harm his client in prison as outside, especially because the allegations are sexual.
The other people who offered sureties were solicitor Geoffrey Shears and Professor Patricia David, as well as a sixth unnamed person.
Speaking after the hearing, Mr Pilger said: "One only has to read the document to understand the enormous service that Julian Assange has put to us and when I say us, I mean the whole of humanity and journalism.
"This is the best type of journalism. It is telling the truth."
Also speaking after the hearing, Mr Stephens said: "We are entitled to an appeal to the High Court and we are also entitled to another day at the magistrates' court.
"At this moment in time, we have not made a final decision."
Mr Stephens added that his colleague discussed the decision with Assange after the hearing.
He said: "Next is further consideration of another bail application."
Asked whether he should be extradited, he replied: "I am convinced the British judicial system is robust enough to deal with this type of case and justice will out."
Mr Stephens said he is sure Assange will be "vindicated".
