Finny does not survive the surgery process to repair his leg (after his fall down the stairs). From the doctor's description, Finny's condition appears to be a blood clot that travels from his broken leg to his heart. This is not an uncommon condition, even with today's medical technology, for patients who suffer leg and foot injuries.
What is ironic is that Gene is told that part of Finny's bone marrow separates from his bone and stops his heart. The irony lies in the cliche about the marrow of life--made famous by Thoreau. It is almost as if Finny has so much "life" inside him that it overflows and takes away his earthly life. The author seems to imply that Finny was not made for this world. He is an idyllic character.