The video obtained by the News of the World shows Harry while still an officer cadet at Sandhurst military academy.
It was filmed in front of other cadets at an airport departure lounge as they waited for a flight to Cyprus to go on manoeuvres.
The newspaper said the prince, who is third in line to the Throne, had called the soldier "our little Paki friend".
Meanwhile, during a faked call to the Queen, as the Commander in Chief of the British Army, the prince says: "Granny I've got to go, send my love to the Corgis and Grandpa."
He finishes saying: "I've got to go, got to go, bye. God Save You ... yeah, that's great."
BBC Royal Correspondent Peter Hunt describes the content of the video
A statement from St James's Palace, regarding the term "Paki", said: "Prince Harry fully understands how offensive this term can be, and is extremely sorry for any offence his words might cause.
"However, on this occasion three years ago, Prince Harry used the term without any malice and as a nickname about a highly popular member of his platoon.
"There is no question that Prince Harry was in any way seeking to insult his friend."
The statement continued: "Prince Harry used the term 'raghead' to mean Taleban or Iraqi insurgent."
A man told BBC Radio Five Live the cadet concerned was his nephew, Ahmed Raza Khan, from Pakistan, who served with Prince Harry at Sandhurst for one year as a Commonwealth cadet.
Iftikhar Raja said his nephew, now a captain in the Pakistani army, would have risen above such terms and had not mentioned the incident to his family.
Mr Raja said: "At no time he told us that he was called Paki or he was a good friend of Prince Harry, I mean, although they served together that is true.
Captain Khan's uncle Iftikhar Raja: 'They should have had more respect for each other'
"But I myself am a British subject, I am proud to be British and if someone called me Pakistani I would be proud to be called that, but Paki is definitely a derogatory remark."
He added: "We expect better from our Royal Family on whom we spend millions and millions of pounds for training and schooling."
Captain Ahmed Raza Khan graduated with Harry from Sandhurst in 2006 receiving a special award from the Queen for being the best overseas officer cadet.
BBC royal correspondent Daniela Relph said this was an extremely embarrassing episode for the prince and the Royal Family.
She said the emergence of the three-year-old video was "unfortunate timing" for Harry, whose image had greatly improved since he served in Afghanistan last year.
"That was a real step up for him, a real sense of maturity that people could see," she said.
She added that as a member of the Royal Family, Prince Harry was held to a certain standard, and everything he said and did was scrutinised "regardless of whether it was banter among colleagues or something that was being used by lots of other people he was working with".
The Army has been trying to recruit soldiers from ethnic minority backgrounds as these are currently under-represented in the services.
A spokeswoman for the Ministry of Defence said: "Neither the Army nor the Armed Forces tolerates inappropriate behaviour in any shape or form.
"The Army takes all allegations of inappropriate behaviour very seriously and all substantive allegations are investigated.
"We are not aware of any complaint having been made by the individual. Bullying and racism are not endemic in the Armed Forces."
A spokeswoman for the Equality and Human Rights Commission said: "These appear to be disturbing allegations and we will be asking the MoD to see the evidence, share that evidence with us and their plans for dealing with it.
"We will then consider what further action might be necessary."
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