The greek

Yes, but there are a great many Greek-owned and operated diners throughout the USA. Practically every diner in NYC today is Greek owned and operated and by those I am referring to typical, American diners. The "Greek diner" is a fairly standardized icon in the American landscape.

I don't know why they chose to remove that aspect of Nick being a Greek immigrant from the original story for this film. It was not a particularly positive portrayal and I'm not sure why it was used in the novel, unless the author was trying to infer some kind of unflattering ethnic stereotype that could have implied a few different things, such as: 1. that Nick was "foreign" and loud and demonstrative in a way that was very different from Frank and Cora and that made the thought of Nick and Cora together intimately as man and wife seem "wrong" or unsavory or offensive 2. that Nick was possessive and chauvinistic toward his wife, Cora, in ways that an American man (in theory) wouldn't be, for example, Nick automatically assuming that Cora would provide care for Nick's paralyzed "half-dead" sister simply because he would tell her to 3. that Nick, being so foreign, was somehow less human and more disposable, easier to do away with?

Maybe they thought it would be needlessly offensive to Greek-Americans to include Nick as a Greek. This was already not a nice story and that would add yet one more detail that is definitely not nice. Nick, in spite of being loud, and getting a little sloppy when he's had a drink or two, is not a bad man, in fact, he's done many kindnesses, certainly for Frank.