What makes answering the question of what makes the eponymous protagonist of Euripides' play Medea a tragic heroine somewhat complicated is that the term "heroine" is being applied anachronistically. When we think of the term "heroine" in popular context in the twenty-first century, we are actually fusing together two different concepts, one having to do with the centrality of the character in the plot and the other having to do with a sense of not only greatness in the sense of importance but moral goodness. In Greek tragedy, these two concepts were not actually linked together. The term hero was used for a race of people who were quasi-divine (Hercules is an example), but this term is not the same as tragic protagonist.
According to the important Greek philosopher Aristotle:
Tragedy, then, is an imitation of an action that is serious, complete, and of a certain magnitude; in language embellished with each kind of artistic ornament, ... Again, Tragedy is the imitation of an action; and an action implies personal agents, who necessarily possess certain distinctive qualities both of character and thought; ...
The leading character in a play was called the "protagonist" (literally meaning "main contestant"). The protagonist was the most important character in the play, and according to Aristotle was expected to suffer reversals, and have experiences that invoked fear and pity in the audience. For this, the character needed to have a certain grandeur, a sense of being larger than life or out of the ordinary, but not necessarily morally good or a role model.
Medea is therefore a strong protagonist, with a certain greatness, strong willed, possessed of magical powers, and part of great historic events. She was also considered by Greek readers to be, like several other important female tragic or epic protagonists (e.g. Helen, Clytemnestra), evil. Thus we can say that she indeed fits the ideal of the tragic protagonist, but not twenty-first century conceptions of a "heroine."
Euripides' Medea proves to be a lesson on the power of both love and revenge. Medea, angered at Jason's marrying of another woman, chooses to enact her revenge upon him. Her love for him proves to be far less important than her need for revenge.
A typical tragic hero, or heroine, possesses hamartia (a tragic flaw) which leads to his or her downfall. This downfall tends to result in an increase in the hero's self-awareness, and the hero normally gains the pity of the audience.
In the case of Medea, some can argue that she does not fit the characteristics of the true tragic heroine. Although she possesses a tragic flaw (pride), she does not increase her self-awareness or gain sympathy from most audiences (given the bloody murder of her own children). The audience is never actually given "closure" to the action of the play--Medea simply drives off with the bodies of her children. Therefore, the audience has no reason to pity her given her "closing" actions.