Does King Lear fit Aristotle's definition of a tragic hero?
Yes, King Lear does fit Aristotle's definition of a tragic hero. Aristotle stated a tragedy must be a drama about persons and things of some importance, where the highly placed hero is brought low through the combination of his or her own faults (the "tragic flaw") and external forces. The situation must be capable of being generalized, and it should induce pity and fear in the viewers. Finally, the drama must end with the attainment of understanding, bringing about a katharsis or "purging" that resolves the pity and fear the audience feels.
King Lear is a highly placed individual, and his proposal to retire from the cares of state is an affair of some importance. His "tragic flaw," the inability to distinguish between sincere and false devotion, is a common fault among people in general, and so audiences easily feel pity and fear when he falls victim to flattery and is ruthlessly betrayed by those he should have been able to trust the most. Finally, Lear approaches a state of understanding at the end of the play, when he realizes (though too late) that Cordelia was in fact the only one of his daughters to be true to him, and for the whole of his life he has been ignorant of the true nature of his role as a king and his own devotion to earthly vanity:
Come, let's away to prison:
We two alone will sing like birds i' the cage:
When thou dost ask me blessing, I'll kneel down,
And ask of thee forgiveness: so we'll live,
And pray, and sing, ... and we'll wear out,
In a walled prison, packs and sects of great ones,
That ebb and flow by the moon. (Act V, Scene 3)
Even though the play ends with the deaths of both Lear and Cordelia, these lines show that before his death, Lear had reached the complete comprehension (anagnorisis), the understanding and acceptance of how he truly fits into the scheme of things, which tragedy demands.