When E. M. Forster came up with the distinction between flat and round characters in his book Aspects of the Novel he defined a flat character as one that is not fully formed or developed and who has maybe only one or two defining characteristics. A round character, by contrast, is one that is fully developed and who is psychologically presented in all of his or her fullness to the reader.
When we consider these definitions, it becomes clear that Holden can only be considered a round character. Consider the way that Salinger, throughout the novel, exposes the full character of this immensely troubled young man through showing the way that he is so unaware of what is happening to him and also how he is almost out of control of his actions. An early section in the novel which is a good example of this is when Holden talks about his brother, Allie, and how he responded to Allie's death:
I slept in the garage the night he died, and I broke all the goddam windows with my fist, just for the hell of it. I even tried to break all the windows on the station wagon we had that summer, but my hand was already broked and everything by that time, and I couldn't do it.
We begin to realise that the narrator who is telling us his story is clearly unreliable as he is not aware how damaged his brother's death has made him, and his rather reckless adventures can be viewed as being a result, at least in part, of his unresolved grief for his brother. Such psychological complexity is only possible in a round character.