This is a really interesting question. First, let us start by reminding ourselves what a tragic hero is. Sophocles is responsible for creating the term based on his Theban plays. A tragic hero is a character who is often a member of royalty, who strives to achieve something and is ultimately defeated. The defeat of the hero my appear to be brought about by forces such as fate or destiny that are beyond his or her control, and the outcome seems almost predetermined. However, often the downfall of the tragic hero is brought about by a fatal weakness or character flaw. In spite of defeat and death, the tragic hero is ennobled by their self-knowledge and wisdom that they gain through what happens to them.
Now let us turn to the father in this unforgettable novel. Although he dies at the end, he is actually successful in what he is aiming for - although he does not live to see it. His son finds a family to join and we are confident that he will be safe and grow up to be an adult in this hopeless world. I would also argue that this story represents the triumph of hope in a hopeless world - there is no sense of the powers of destiny exerting their strength and treating humans like puppets. From the very start, we are presented with the father who has determined not to give in and die like his wife, however hopeless the situation, but wants to keep on struggling and fighting for survival. Individual freedom is shown to be stronger than destiny in this novel. Therefore I don't know that it is correct to say that the father suffers a tragic "downfall." If he has a flaw, it is the way that he has almost lost faith in the goodness of human nature, that interestingly his son strives to reignite within him. His death comes as a result not of his tragic flaw but because of his desire to save and protect his son.
Therefore, personally, whilst I think the death of the father is definitely sad, we cannot say that he is a tragic hero according to the original definition of the word - he is successful and his death is a triumph of individual free will and the desperate urge to keep on living no matter what the circumstances.