The angry diamond is some man's wife. They are at Gatsby's party. As Nick observes, the wife tried to remain calm and dignified while her husband was obviously flirting with some actress. She gets fed up, loses her cool (who could blame her) and starts attacking him. I suppose "you promised" means 'you promised not to flirt with other women.' This seems like a promise any self-respecting man could keep, but this guy is a player. And although the wife is the one scorned, the fact that she is at Gatsby's party implies that she's like the husband, actress, and all the other shallow party-goers. Further, she may even be with this guy for social standing. Nick's impression of 99% of Gatsby's crowd is that they all use him for his social status and to party. They're all glitz, glamour and no substance: a diamond is a symbol of wealth, status, but it's just a rock. She is like an angry diamond because she is angry and, from Nick's perspective, as superficial as the rest of them. Although she is being disrespected, calling her an "angry diamond" reduces her to an object: she's just for show - this could be her own doing - she's there to show herself, part of the "in" crowd, and/or her husband is there to show her off for whatever reason. A bit misogynistic maybe, singling her out. I'd want the actress and the husband dissed as well: maybe call them the brainless ruby and the thoughtless piece of slate.
The "angry diamond" simile is very much tied to the fact that a diamond is a promise of fidelity. The woman who says this to her husband is alluding to the fact that he promised to be faithful, yet he is flirting with the available women at the party. By the way, the new Baz Luhrman movie highlights this line at the underground speakeasy with the tablecard reading "The Angry Diamond" as the name of the club...interesting tie to the simile and the fact that most of the men at this establishment are cheating on their wives...
In Ch.3 Nick Carraway attends a party at Jay Gatsby's house. Initially, as soon as he arrives at the house he is completely overwhelmed by the lavish splendor and extravagance of the party:
On buffet tables, garnished with glistening hors-d'oeuvre, spiced baked hams crowded against salads of harlequin designs and pastry pigs and turkeys bewitched to a dark gold.
A little later as the party warms up, he views the anonymous interpersonal interactions of the party guests very cynically:
The bar is in full swing, and floating rounds of cocktails permeate the garden outside, until the air is alive with chatter and laughter, and casual innuendo and introductions forgotten on the spot, and enthusiastic meetings between women who never knew each other's names.
At about two in the next day Nick notices that the men are reluctant to return home with their wives. They wish to continue flirting with the other beautiful women:
I looked around. Most of the remaining women were now having fights with men said to be their husbands. Even Jordan's party, the quartet from East Egg, were rent asunder by dissension. One of the men was talking with curious intensity to a young actress, and his wife, after attempting to laugh at the situation in a dignified and indifferent way, broke down entirely and resorted to flank attacks--at intervals she appeared suddenly at his side like an angry diamond, and hissed: "You promised!" into his ear.
The wife's anger and rage are expressed in her eyes which glint and sparkle like a diamond and she hisses her displeasure to her husband like a snake saying that he promised to behave and that he has been promising her perhaps for a very long time to take her home.
One of the most important characteristics of modern fiction is that the prose is poetic. Here Fitzgerald very poetically uses a simile to describe the eyes of the wife glinting suggestively with rage to that of a sparkling diamond. At the same time the simile itself is an instance of 'pathetic fallacy' for the inanimate diamond is endowed with the human quality of anger.
Ruskin who coined the term 'Pathetic Fallacy' in his "Modern Painters" (1856) defined it as “to signify any description of inanimate natural objects that ascribes to them human capabilities, sensations, and emotions."