No, according to the classic definition, mainly because the main characters are not of high rank and therefore cannot “fall from a high place.” The correct term might be “melodrama,” if you consider the characters tugging at the audience’s emotions with exaggerated expressions of grief. The other problem with the term “Tragedy” is that what happens to the characters is not “tragic” that is, irreversibly destructive (usually this means death.) Blanche is presumably going to a hospital, Stella and Stanley will continue with their damaged relationship; and none of them undergoes any sort of catharthis or cleansing. The most obvious absence for a definition of tragedy is the lack of awareness by the characters – there is little or no realization of what just happened, and therefore no resolution. The 20th century has usurped the word “tragedy” and softened it to mean “not happy.” One valuable classic definition of comedy is “tragedy avoided.” The term “tragicomedy” has also been used, to mean something “too bad, but ironic at a distance.” So the warning here is: don’t assign genre names too strictly, because they are tools for post-creation critics, not blueprints for artists.
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As we are no more in the Elizabethan Era where the notion of tragedy was restricted, it might be interesting to consider the wider implications of the tragic situations our characters face. Blanche is a tragic heroine who faces her moral death in the play; she lives a life in death ever since her young degenerate husband commits suicide on account of Blanche's insensitivity towards his homosexuality. There is also Stella who has to keep on living with his husband even when knowing that Stanley might have raped Blanche but as she is economically dependent on the man, she has no other choice with a new-born baby to raise.
Personally, I would say that the end is really very tragic. The way that Blanche is taken to an asylum, Mitch's crying, Stanley's nonchalence and Stella's sobbing and crying for Blanche without the latter even looking at her is all very sublime. Depend upon it, catharsis can be surely seen here. ;)