In a statement, WikiLeaks said Assange's health had deteriorated during the seven years he spent inside the Ecuadorian embassy in London "under conditions that were incompatible with basic human rights".
It said the UN had twice found him to be arbitrarily detained and asked the UK to honour its commitments to international law and free him.
"During the seven weeks in Belmarsh, his health has continued to deteriorate and he has dramatically lost weight," the statement said. "The decision of prison authorities to move him to the health ward speaks for itself."
He was later sentenced to 50 weeks in jail for skipping bail at the time he fled into the Ecuador embassy; at that time, he was wanted in Sweden for questioning on rape allegations made against him by two women. Sweden has subsequently reopened the rape probe and sought his extradition; it had dropped the charge and cancelled the warrant in May 2017.
Soon after his arrest, the US sought his extradition to face charges of computer hacking and being involved in a compromise of classified information, a charge that carries a maximum sentence of five years.
Washington revealed its hand on 24 May when it hit Assange with 17 new charges under the US Espionage Act over his alleged role in leaking documents from former US intelligence analyst Chelsea Manning in 2010.
WikiLeaks editor-in-chief Kristinn Hrafnsson said: "Julian's case is of major historic significance. It will be remembered as the worst attack on press freedom in our time.
"The people need to voice their condemnation; it is theur politicians, their courts, their police and their prisons that are being abused in order to leave this black stain on our history. Please act now to avert this shame."
On Thursday UK time (Friday AEST) a formal hearing will take place in Westminster Court on the US extradition request.