The elements of the story that seem to fit the tragic mold: 1) the man's strength leads him to failure. 2) a glorious feat is rendered merely symbolic because nature steals the glory.
These elements characterize some formal tragedies in the Greek tradition, but, in agreement with the posts above, they don't necessarily qualify The Old Man and the Sea as a tragedy.
Isn't all life a tragedy when we all eventually die? Therefore, The Old Man and the Sea is, in part the tragic end of the essence of Santiago as a fisherman. For, he has been defeated by Nature in the form of the sharks who eat the Marlin. Without the big fish to sell, Santiago has nothing--neither physical sustenance nor spiritual sustaenance as his life as a respected fisherman is ended, symbolized by his telling Manolin to tell Pedrico to break the skiff up and use the wood in fish cages.
I agree that some of what happens in this story may be tragic but I would not call The Old Man in the Sea a tragedy. It is a tragic loss for Manolin that he is not allowed to fish with the old man. Santiago lives in dramatically poor circumstances, and that is tragic. Fishing has been awful for the old man, and that is a kind of tragedy for a man who makes his living at it. We feel like this is a lonely, weak old man--until he conquers every adversity, brings in the huge marlin, and battles with sharks. Santiago is a testament to perseverance, and that's what keeps the story from being a tragedy.
It's not a tradgedy in the classical sense. It does not have a true hero. It's basically a man and a fish. All of the action occurs on a different level. The tragic point is the battle going on inside the head. I find it sad, but I would not say it's tragic.
It is going to be difficult to make Hemingway's work a tragedy. The very idea of Santiago representing one of the quintessential Hemingway Heroes would preclude a sense of tragedy in it. If one tried to paint this as a tragedy, it would have to lie in the very idea that Santiago is poised against society. Santiago's condition of loneliness is tragic. Grieving for his wife, years after his death, Santiago's life is one where his state of being is seen as unlucky and unfortunate. The fact that he identifies with Dimaggio is another example of a tragic condition. Santiago loves the fact that Dimaggio fights through pain and endures unspeakable physical and emotional pain in giving everything in being "Joltin' Joe Dimaggio." Santiago could have identified with Gehrig or Ruth, figures of triumph or monolitic admiration. Yet, he chooses Dimaggio because of the melancholy quality that is intrinsic to the Yankees legend, something that he sees in his own state of being in the world. Santiago's condition of respecting nature and yet killing it to live is something of tragic conditions. When Santiago discusses the pain of killing a marlin while its mate watched the ordeal, one sees the tragedy in both the situation and how Santiago identifies with it. Santiago is a character that will never receive validation, perhaps his most hauntingly tragic condition. There will never be the type of triumph and sense of totality that he might deserve or that he might want. In the end, Santiago recognizes this, but has no other choice than to persevere with his desire to fish, to live, and to dream what no one else can. In this, Santiago is both tragic and human, a connection that Hemingway might be saying is inseparable, like the marlin and its mate.