The singer labelled Prince William and Kate "arrogant" in the aftermath of the King Edward Hospital nurse's suspected suicide.
Nurse Jacintha Saldanha was found dead last week after transferring a call in which Australian DJs Mel Greig and Michael Christian pretended to be Queen Elizabeth II and Prince Charles to gain medical details about the hospitalised Duchess of Cambridge.
The former Smiths frontman called for the royal couple to show remorse for the incident, and that the blame should not be solely placed on the DJs.
He told New Zealand's 3News: "Even with the recent story about the nurse killing herself in King Edward Hospital, there's no blame placed on Kate Middleton, who was in the hospital as far as I could see for absolutely no reason.
"She feels no shame about the death of this woman, she's saying nothing about the death of this poor woman. The arrogance of the British royals is staggering, absolutely staggering. And why it's allowed to be I really don't know."
He added that the duchess's reported morning sickness was not worth a hospital visit, saying: "Does she have a health condition...? What is the health condition?
"I mean morning sickness already? So much hoo haw and then suddenly as bright as a button as soon as this poor woman dies she's out of hospital? It doesn't ring true.
"And I'm sure the Palace and Clarence House put maximum pressure on this poor receptionist and nurse, and of course that's kept away from the press.
"I'm sure the British press hounded this poor woman to her death, that's kept away. And by this time next week she'll be forgotten. And that's how the British royals work."
[Michael Christian and Mel Greig on air at 2DayFM]
Australian radio station 2Day FM have announced that they will donate ad revenue to the family of Jacintha Saldanha.
Morrissey has criticised the British royal family for many years, recently describing Prince Harry as "an idiot".
He caused controversy earlier this year by comparing the London 2012 Olympics to Nazi Germany.