In a tragedy, the heroine or hero usually has a character flaw that leads to their downfall. Is this true in Romeo and Juliet? Why?

Yes, it is certainly true in Shakespeare's Romeo and Juliet that the hero and heroine have a character flaw that leads to their respective downfalls!

It is traditionally argued that Romeo's tragic flaw is his impulsivity. He is impulsive in love (falling madly in love with Rosaline and then shifting those feelings to Juliet; later marrying Juliet after knowing her for less than a day), in his social life (choosing to fight and kill Tybalt rather than stand down after Mercutio is murdered), and in his own life (choosing to kill himself after learning of Juliet's "death"). 

Thus, Juliet's tragic flaw might be her naive loyalty to Romeo. Juliet repeatedly places her trust and faith in Romeo, who--as we can see above--frequently makes poor and/or rash choices and is overly-engaged in the conflict between the Montagues and Capulets. Juliet trusted Romeo to be a responsible husband and ally, and was, therefore, quick to agree to marry him; Romeo betrayed that trust when he killed Juliet's cousin, Tybalt, resulting in his banishment from Verona... and ultimately, Juliet's desperate ploy to save her marriage by faking her own death.