In a letter to the Australian radio station 2day FM's parent company, Lord Glenarthur, chairman of King Edward VII's Hospital, said he wanted to "protest" against the "extremely foolish" gag.
His comments follow the apparent suicide of nurse Jacintha Saldanha, 46, yesterday morning.
Lord Glenarthur said her death was "tragic beyond words".
The letter to Southern Cross Austereo said the immediate consequence of the station's "premeditated and ill-considered actions" led to the "humiliation" of Ms Saldanha and another nurse.
"I appreciate that you cannot undo the damage which has been done but I would urge you to take steps to ensure that such an incident could never be repeated," it added.
A spokesman for the private London hospital said the letter was sent earlier today.
Ms Saldanha's devastated family were being comforted by relatives and friends at their terrace home at Greystoke Avenue, Southmead, Bristol.
A friend at the address said Ms Saldanha's partner Benedict Barboza, 49, and their teenage son and daughter, aged 14 and 16, were "very, very shocked and unhappy at the tragedy".
The devoted nurse and mother-of-two was pronounced dead at staff accommodation close to the hospital where the pregnant Duchess had been treated for a severe form of morning sickness.
News of her death led to a huge backlash against the two presenters, Mel Greig and Michael Christian, who broadcast the prank on their show on 2day FM.
They have now been taken off the air while the Sydney-based station has been inundated with complaints.
Neighbours in Bristol appeared torn by anger at the DJs combined with shock and sadness for the family.
Some questioned whether there may have been more pressing concerns, unknown as yet, on the mind of the nurse.
"She was a lovely, lovely person who always spoke to you when you saw her in the street," said neighbour Mary Atwell, 56.
"The kids were always polite and well-behaved. The boy often played football on the green outside with his friends. I saw him go off to school yesterday.
"Later in the afternoon I saw a police car outside their home but I didn't think much of it. I suppose that was when they heard.
"God knows how they must be feeling now. There is a lot of family going in there today.
"She must have taken the whole thing very badly because I don't think the hospital disciplined her for what happened.She simply couldn't live with herself I suppose.
"But you could always see that she was very dedicated to her job. She was a lovely person."
She said that even when she was away from work she was kind to neighbours and helped them.
"She used to walk an elderly neighbour who has dementia, who lives a few doors along, down to the shops and back.
"She fitted in well around here, they all did. They've lived here for at least 10 years and were very quiet and pleasant.
"I didn't realise she was working in London. She must have been travelling up there regularly. That shows you how dedicated she was, a good person."
She also hit out at DJs Greig and Christian, who put on fake English accents and pretended to be the Queen and Prince of Wales for the prank.
Ms Atwell said: "Both DJs should be sacked they should never have been allowed to do what they did.
"They are definitely responsible for this tragedy. She would be alive today if they hadn't have made that call. It should never have happened."
A single bouquet of flowers wrapped in pink paper, propped up against the home's iron gates, is the only outward sign of yesterday's tragic events.
Other neighbours were equally angry about the hoax call. "Quite honestly these Australian DJs should be sacked," said Alf Prideaux, 82.
"I think that people like them should step back and consider what they are doing and take note of the results of their mockery.
"They picked the wrong time and the wrong person. You never know who you'll get on the other end of the line.
"I don't think they thought through the possible consequences of what they did."
He said that his generation retained a respect for authority and did not find mockery clever.
He spoke as he left Bethany Southmead Baptist Church, opposite the nurse's family home, where he is one of the congregation.
He said it was likely the church pastor would say prayers for Ms Saldanha and her family when they meet tomorrow.
"I don't know if he knows she is from here. But knowing the pastor it is the sort of thing he would do."
A female pensioner, who did not want to be named, said: "I came down here last night walking the dog and saw all the commotion and somebody told me what had happened.
"It was such a terrible shock. Tragedy like this seems to be the way of the world now.
"She must have been a very special person. I think she was a very sensitive person who found herself in the spotlight without the support of her family.
"Sometimes, with Christmas coming, it can be very depressing being alone as well. I never look forward to Christmas."
Dudley Todd, 79, who lives nearby at Coldbeck Close, said: "I can't help thinking there is more to this.
"It just doesn't make sense when you have two children depending on you.
"It all seems a bit odd because it was not that serious. Maybe the police have discovered something they are not yet telling us about."
In a statement last night, Ms Saldanha's family said they were "deeply saddened" by the death and asked for privacy.
They said: "We as a family are deeply saddened by the loss of our beloved Jacintha. We would ask that the media respect our privacy at this difficult time."
Ms Saldanha answered the presenters' call and, believing they were members of the Royal Family, put them through to another nurse who described Kate's condition in detail.
But Southern Cross Austereo chief executive Rhys Holleran stood by the two DJs and said they were shocked and devastated by news of Ms Saldanha's death.
At a news conference in Melbourne, he said: "This is a tragic event that could not have been reasonably foreseen and we're deeply saddened by it.
"I spoke to both presenters early this morning and it's fair to say they're completely shattered."
Mr Holleran said the pair had been offered counselling, adding: "These people aren't machines, they're human beings. We're all affected by this."
He would not say who came up with the idea for the call, only that "these things are often done collaboratively".
He said he was confident the station had not broken any laws, noting that prank calls in radio have been happening "for decades".
"They're not just part of one radio station or one network or one country - they're done worldwide," he said.
The company said that, by mutual consent, the hosts would not be returning to their show until further notice.
It said in a statement: "Southern Cross Austereo (SCA) and 2Day FM are deeply saddened by the tragic news of the death of nurse Jacintha Saldanha from King Edward VII's Hospital and we extend our deepest sympathies to her family and all that have been affected by this situation around the world.
"Chief executive officer Rhys Holleran has spoken with the presenters. They are both deeply shocked and at this time we have agreed that they will not comment about the circumstances.
"SCA 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."
In their initial apology the two presenters said: "We were very surprised that our call was put through. We thought we'd be hung up on as soon as they heard our terrible accents.
"We're very sorry if we've caused any issues and we're glad to hear that Kate is doing well."
But the pair, who broadcast the call on their Hot30 Countdown show, continued to boast about the stunt, describing it as the "easiest prank call ever made".
A flood of complaints have been made to the Australian Communications and Media Authority, which regulates radio broadcasting.
The Duke and Duchess of Cambridge last night sent their condolences to Ms Saldanha's family.
In a statement, St James's Palace said: "The Duke and Duchess of Cambridge are deeply saddened to learn of the death of Jacintha Saldanha.
"Their Royal Highnesses were looked after so wonderfully well at all times by everybody at King Edward VII's Hospital, and their thoughts and prayers are with Jacintha Saldanha's family, friends and colleagues at this very sad time."
The spokesman stressed that they had not complained to the hospital about the hoax call, saying: "On the contrary, we offered our full and heartfelt support to the nurses involved and hospital staff at all times."
A post-mortem examination will take place next week and the death is not being treated as suspicious, Scotland Yard said.