Santiago says nothing beat him except that he went too far out. Hemingway's message is that his protagonist is destroyed but not defeated.Do you...

Wow.  You  have a lot going on here for such a short
story.  Destroyed and defeated seem to be the
essential words in The Old Man and the Sea, and they probably hold
the key to answering your own question--is Santiago a tragic hero.  When I look at the
general criterion for a tragic here, Santiago does not quite fit.  Almost, but not
quite.  One, he does suffer his loss, at least in part. by his own doing.  Santiago goes
out alone and he goes out too far.  There is a natural consequence of bringing is such a
fish alone so far away from land and other fishermen.  He could have cut the line and
cut his losses and been better off than he was in the end.  Two, he does not seem to
have the thing commonly referred to as a fatal flaw, at least not to the degree most
other tragic heroes have it.  His one glaring show of pride in the midst of his
amazingly humble life is his pride in being a good fisherman.  This is a man who allows
Manolin to provide him fresh baits, who owns next to nothing, who wouldn't eat much of
the time if it were not for the boy.  Yet, he is convinced he is still a good enough and
strong enough fisherman to catch this big fish.

Third,
a tragic hero generally loses more than he should--his punishment exceeds his crime. 
Not so here.  Santiago's hands and body are worse for the wear, for sure, but he has
literally lost nothing more than a few days of fishing.  It's true that this incident
undoubtedly broke his spirit (thus the reference to destroy);
however, we understand Santiago will go on to fish another day.  He has not been
defeated.  Most tragic heroes do not have that luxury; the consequences of their actions
generally keep them from participating in the life of their choosing again.  Often, the
punishment, the price they pay, is death.  Not so for Santiago.  Fourth, does he learn
from his mistake, as most tragic heroes admit to doing--often right before they die. 
I'll leave this one for you to decide. Finally, the audience must feel pity and awe. 
That we certainly do.

In short, then, Santiago is perhaps a
kind of tragic hero, one who is, indeed, destroyed but not defeated.  According to the
traditional definition of the term, though, he does not fit the entire description. 
That doesn't make his loss any less profound, though, it seems to me.  Rather, it makes
him a relatable figure who battled and lost but will live to battle
again.

An interesting thing to think--and
write--about!