Jacintha Saldanha answered the phone when presenters from Sydney's 2Day FM called pretending to be Queen Elizabeth II and William's father Prince Charles, before passing it onto a colleague who divulged details of Kate's condition.
Saldanha was found dead on Friday, with police saying her death was not being treated as suspicious. Her employers, London's private King Edward VII hospital, refused to comment on media reports that she had taken her own life.
The radio station said Saturday the presenters, Mel Greig and Michael Christian, "are both deeply shocked" over news of the death.
The station and the hosts "have decided that they will not return to their radio show until further notice out of respect for what can only be described as a tragedy", 2Day FM and its owners Southern Cross Austereo said in a statement.
"We extend our deepest sympathies to her family and all that have been affected by this situation around the world," it said in the statement.
Soon after news of the death broke, the radio station's Facebook page was bombarded with thousands of comments, many attacking the presenters and calling for them to be sacked.
"Not so darn funny now is it? A British nurse is DEAD for the sake of a couple of cheap laughs. Shame on you!" wrote Kim Wilson.
More than 11,400 comments had hit the 2Day FM page by noon Saturday, some saying Greig and Christian had blood on their hands and calling angrily for them to be dismissed.
"Hope you get your comeuppance and are looking long and hard at your actions and their consequences. Hope you're both sacked and spend a very long long time looking for work," wrote Alastair Drake Hardwick.
For some the incident had echoes of Prince William's mother Diana, Princess of Wales, who died in a car crash in Paris in 1997 while being pursued by paparazzi.
"One would think... the death of Princess Diana would have taught the media a lesson about invasion of privacy of the royal family, but I guess not," said one commentator posting as Lora LB.
The news prompted wide coverage in Australia, where the prank initially divided the public, with some seeing it as "a bit of harmless fun" and others saying a line had been crossed and everyone had a right to privacy.
Australia's Prime Minister Julia Gillard said the death was a terrible tragedy.
But there was also support for the presenters, with Jeff Kennett, chairman of the national depression initiative Beyondblue, saying the pair had not intended to cause harm and he hoped the public would support them.
Greig and Christian apologised after an uproar in Britain over the hoax but the station initially milked the publicity as the "biggest royal prank ever".
It has now removed the link to the hoax from its website and the presenters' Twitter accounts have been suspended.
The radio station has been in the public eye in the past, notably in 2009 when two other presenters ran a segment in which a 14-year-old girl was given a lie detector test in which she revealed she had been raped.
Communications Minister Stephen Conroy said the Australian Communications and Media Authority had received complaints about the hoax call and the regulator would now decide whether to investigate for any breach of codes of practice.