Ilych's thought process and his descent into memory are the best evidence for the moral judgement being placed on his life.  As he weakens, Ilych longs for the happy life that he once lived.  But given the time to focus on it, he understands that while his childhood was happy, his adult life has been a pursuit of goals and materialistic recognition, and has been devoid of happiness.  The fact that he is actually not able to die until he has found sympathy for his family reinforces the need for him to undergo spiritual enlightenment.  He has not lived his life well; he has wasted it on the trivial, and he must understand that in order to let go of the pain and be at peace.

Incidentally, Ilych's injury is symbolic of this judgment.  Hurting himself as his admires his "achievement" of a respectable home proves that this goal was one of moral corruption.