Not being a royalist I am almost reluctant to weigh in on the subject of Prince Harry using the nickname "Paki" in referring to a fellow soldier. But my point is not about royalty as much as the absurd degree to which the media pick up on and abuse language.
The term Paki is not abusive to everyone - listen to South Asians speaking and it becomes apparent that a person from Pakistan is a Paki just like someone from the province of Baluchistan (in Pakistan) is called a Baluchi. Following the logic of the British media and others, someone from this area of the world can be a Baluchi but not a Paki.
I would guess that the nickname "Paki" has an unfortunate etymological history in the media's coining of the term "Paki-bashing" several decades ago when racial tensions in Britain's cities were high. If so, then Prince Harry is actually being judged by the very originators of the "racist language" he is accused of using.
A case of the pot calling the kettle . . . .