I believe that it does. Tragedy, as created by the ancient Greeks and as explored by Shakespeare many centuries later, develops in relation to their protagonists in drama. The leading character in a tragedy is a person of high social station, someone who is elevated in society. These characters are placed in situations in which they confront forces beyond their control, and their lives subsequently are destroyed.
Both the Greeks and Shakespeare placed their characters in situations that could not be resolved through intelligence and human reasoning. Their destruction assumes tragic proportions, however, because they are destroyed as a result of some flaw within their own characters, often a part of themselves they fail to recognize until it is too late.
The protagonist in Faulkner's story is Emily Grierson, an aristocratic woman from a prominent Southern family who is trapped in the Southern culture long after the South has fallen. Miss Emily cannot be considered a tragic heroine in the classical sense, not completely, because Faulkner's subtle story introduces the idea that among her many problems lies mental illness that she had inherited from her family, and that is surely not a flaw in her character as a human being.
However, other tragic elements are found in the story. Miss Emily occupies an elevated social rank in Jefferson because of her family name; she is not of the "common people" in her post-Civil War community. She is set apart from them by her exalted Southern family history.
She is also a woman who is placed in circumstances beyond her control; her domineering father controls her throughout his lifetime, robbing her of her own life, demanding her constant attention and preventing her from marrying by destroying any courtship she might have had with a young suitor. Also, the strict Southern social conventions of the society in Jefferson dictate that she live her life in a "respectable" manner required by her family name.
After her father's death, Miss Emily confronts the forces of her society, rebels against tradition, and chooses to have a socially unacceptable love affair with a Yankee construction foreman, Homer Barron. Her defiance is public and blatant, but soon Emily's rebellion is crushed. Townspeople in Jefferson call in Emily's relatives to force her to conform in living her life. In the ensuing days, insanity overtakes Emily Grierson, which is made clear in the story's shocking conclusion.
The story is not a classical tragedy, but certain tragic elements exist in it. High-born Emily Grierson confronts forces beyond her control, the forces of Southern tradition and the judgmental nature of her community. She cannot use intelligence to reason her way to a solution of her conflicts, and she is ultimately destroyed.
If Miss Emily has one "fatal flaw" in her own character, perhaps it is lack of courage as a young woman to defy her father and make a life for herself, but her will was weaker than his. Like Shakespeare's five tragic heroes and like Sophocles' tragic hero, Creon, in Antigone, Emily struggles against the forces she confronts and tries to overcome her fate, but it is too late.