Why did Ivan stop screaming?
Tolstoy believes that we can only truly lead meaningful lives if we gain a deep, spiritual understanding of that mortality which is our common human fate. Yet as the story begins, Ivan has no understanding of this. He's led a crushingly inauthentic life, one characterized by materialism, self-interest, and an obsession with the fleeting moment, the here and now. In other words, he has no true sense of his own mortality; he acts like he's going to live forever.
Ivan's spiritual life is nonexistent. And when he begins to experience the pain and torment of his physical ailments, he has nothing to fall back on; he cannot transcend the intense pain and the misery they cause. But eventually wisdom comes to Ivan Ilyich, and the screaming stops. He realizes that every single one of us is a synthesis of the material and the spiritual. He has spent the whole of his life up to this point living a purely material existence. Now, at long last, he is rediscovering the spirit within. And this rediscovery floods his heart with joy, a spiritual joy that conquers physical pain.
Ivan Ilyich's death ultimately comes to be seen as a regaining of lost spirituality rather than the simple expiration of his last breath. It has a qualitative dimension, one dependent on our constitution as not just physical creatures in a world of nature, but as profoundly spiritual beings whose deaths ultimately matter.
Ivan's course to death is a long and tedious one. In his last three days he screams day and night. It began as the words "I won't!" and became fixed on the "O." In Chapter 11, Ivan realizes that all he lived for was "falseness and deception." With this realization came a new "grinding and suffocating" pain, which is what precipitated his screaming.
Then, "some force struck him in the chest and side" and he fell through the black bag he had been struggling against and saw a light. Simultaneously with this, and in some ways (perhaps in all ways) precipitating this, Ivan was flailing his arms; his small son came; caught one of Ivan's flailing hands and kissed it and began to weep. At this moment Ivan see the light and realizes that his life, which looked like it was right and good, was in fact not the right thing.
This realization, coming at the moment his son kissed his hand and wept, made him ask "What is the right thing?" It is this question, a response to his epiphany of revelation, to which he listened for an answer that caused him to stop screaming and be quiet. Ivan felt the boy's kiss and tears; he saw that he had not lived rightly; he knew it could still be remedied; he listened to hear the answer to what the right thing is; and he, while listening for the way to proceed, quit screaming.