The first step in this question is to define the characteristics of a tragic hero. Generally we define a tragic hero as someone who is great, who dies, and whose death is his own fault brought on by a tragic flaw. Specifically, Aristotle (who defines the characteristics of the tragic hero first in Greek tragedies) was clear that the man must be a hero, but that...
...the hero's misfortune is not brought about 'by vice and depravity but by some error of judgment.'
In looking at J.D. Salinger's The Catcher in the Rye, the first difficulty with Aristotle's definition is that Holden Caulfield does not die. A tragic hero, by Aristotle's definition, must die. Holden also is not a great man, not a hero. He "talks a good game." He is greatly concerned about the innocence of children as he alludes to the Robert Burns poem:
"You know that song, 'If a body catch a body comin' through the rye'? I'd like—"
"It's 'If a body meet a body coming through the rye!'" old Phoebe said. "It's a poem. By Robert Burns."
"I know it's a poem by Robert Burns."
She was right, though. It is "If a body meet a body coming through the rye." I didn't know it then, though.
"I thought it was 'If a body catch a body.'" I said. "Anyway, I keep picturing all these little kids playing some game in this big field of rye and all. Thousands of little kids, and nobody's around—nobody big, I mean—except me. And I'm standing on the edge of some crazy cliff. What I have to do, I have to catch everybody if they start to go over the cliff—I mean if they're running and they don't look where they're going. I have to come out from somewhere and catch them. That's all I'd do all day. I'd be the catcher in the rye and all. I know it's crazy, but that's the only thing I'd really like to be. I know it's crazy."
This passage speaks to both Holden's lack of greatness and his flaw. Holden may be concerned for the innocence of children, but it is unrealistic that he could save the children running through the fields of rye because Holden is such a child himself. He is intuitive enough to be able to read phonies such as Stradlater, but the biggest phony is Holden himself. He may look older than his age, but he acts like an adolescent and is then critical of others who do the same thing, as if they are to be faulted for doing what he does himself. For example, if Holden was so adult, he would find better ways of dealing with his frustrations. When Stradlater goes out with Jane Gallagher, an friend (an old "crush") of Holden's, Holden gets very nervous and resentful. When Stradlater returns later, Holden decides to pick a fight with him—he starts off by doing his best to annoy him:
You weren't allowed to smoke in the dorm, but you could do it late at night when everybody was asleep or out and nobody could smell the smoke. Besides, I did it to annoy Stradlater.
Holden may think himself as an adult, but the other characters in the book treat him in such a way that we can see he is not perceived by others as an adult. In terms of his flaws, it would probably be that he has many of the same faults that he hates in others:
...despite Holden's strong disdain for phony qualities, he exhibits some of the qualities that he abhors...
It may be, then, that Holden seems a lost, lonely and tragic figure. He is not, however, a tragic hero.