Ms Saldanha unknowingly transferred the call to the ward of Prince William's wife Catherine, where a second nurse gave the DJs confidential information about the Duchess, who was being treated for acute morning sickness.
Mr Barboza has called for a thorough investigation into the death of his wife.
Speaking on the family's behalf and with them by his side, British Labour MP Keith Vaz said: "We need to know the facts, fully and clearly. There are unexplained circumstances. The family were in the dark."
Radio station 2Day FM presenters and management, along with hospital representatives and the Duke and Duchess of Cambridge, have conveyed sympathy over Ms Saldanha's death.
Meanwhile a squabble continues as 2Day FM management insist they made numerous attempts to contact the hospital before going to air with the prank call. The hospital denies the claim.
A post-mortem examination was conducted on Tuesday and an inquest into Ms Saldanha's death is likely to be opened and adjourned this week.
* If it's an emergency and you feel like you or someone else is at risk, call 111. Or call Youthline 0800 376 633, Lifeline 0800 543 354, Depression Helpline 0800 111 757, What's Up 0800 942 8787 (noon-midnight).
- AAP