Self-avowed republican Mr Rajan, who presented the controversial new BBC documentary The Princes and The Press, made the comments in articles written for the Independent in 2012. The remarks were made before he joined the BBC and he has today apologised.
In an open letter to the Duke and Duchess of Cambridge after they announced they were expecting their first child in December 2012, Mr Rajan called on the couple to “renounce the luxuries of royal patronage and aristocracy” as he described their public role as a “total fraud”.
Mr Rajan said: “Neither of you have a special claim on the glorious city of Cambridge, so quit pretending that you do.”
In another column on the Diamond Jubilee in January 2012, Mr Rajan accused the Duke of Edinburgh of being a “racist buffoon” and described Prince Charles as “scientifically illiterate”.
He added of the royals: “Aside from the Queen – whose public image is crafted by an ever-expanding team of propagandists – this clan is unusually full of fools.”
Mr Rajan’s past comments were highlighted by the Daily Mail’s Richard Eden after The Princes and the Press sparked controversy.
The second instalment on the two-part programme presented by Mr Rajan aired on BBC Two on Tuesday.
The documentary explored the relationship between Prince William, Prince Harry and the media.
But the show prompted a furious reaction from Buckingham Palace, Clarence House and Kensington Palace.
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“I wrote things that were rude and immature and I look back on them now with real embarrassment, and ask myself what I was thinking, frankly.
“I would like to say sorry for any offence they caused then or now.
“I’m completely committed to impartiality and hope our recent programmes can be judged on their merits.”
A BBC spokesman said: “This article predates Amol’s work at the BBC.
“Once journalists join the BBC, they leave past views at the door.
“Amol is an experienced BBC journalist who reports on all of the topics he covers in an impartial way and in line with the BBC’s editorial guidelines.
“All BBC current affairs output is required to be impartial.”