I'm of two minds about this. On the one hand, of course using racial epithets is wrong. But on the other, in combat a certain amount of dehumanization of the enemy is to be expected and is perhaps even a necessary survival mechanism for some. Like it or not, your goal is to kill other human beings for some purpose, whether rational or irrational. To do so day in and day out requires you to think of your enemy in abstract or insulting terms, lest you start to sympathize with them to the point that you are consumed with guilt over killing them. This is not to say that you don't have to recognize points where dehumanizing the enemy crosses the line - such as torturing or killing the enemy when capture is a viable option.
It's a sad testimony to the harsh reality of war, and certainly not an ideal situation by any means, but I don't know if I would go so far as to equate it to racism as you would encounter in normal peacetime society. At least I wouldn't put it on the same level of "badness." I don't think everyone who used Kraut or Jap or Gook during time of war is necessarily a bigot. If they continue to do so well into peacetime and against people who are not their enemies, then yes, they are. But if you labeled every soldier who ever used racial epithets or insulting names aagainst the enemy a racist or bigot, the almost all of them would probably be racists or bogots. The same for using those names against your friends as a joke.
Plus, if the press started scrutinizing other members of the military to the same degree as they do Harry, I think they would be fairly shocked if this is considered a terrible example. Granted, as a member of the royal family he should be a little more used to it at this point and be more conscious of what he's saying, but I do still feel a twinge of pity for him. He's just trying to fit in and be one of the guys, and no matter what he can't.