What is implied by Tolstoy's calling them not friends, but "nearest acquaintances"?
In "The Death of Ivan Ilyich," prior to being sick and near-death, Ivan had only acquaintances, not friends. Acquaintances are formal, but friends are intimate. Acquaintances are bothered by death; friends are tolerant. Acquaintances go through the motions of mourning; friends actually grieve.
Before he dies, he will have made one friend, Gerasim.
Gerasim did it all easily, willingly, simply, and with a good nature that touched Ivan Ilych. Health, strength, and vitality in other people were offensive to him, but Gerasim's strength and vitality did not mortify but soothed him.
Only Gerasim recognized it and pitied him. And so Ivan Ilych felt at ease only with him.
Gerasim alone did not lie; everything showed that he alone understood the facts of the case and did not consider it necessary to disguise them, but simply felt sorry for his emaciated and enfeebled master. Once when Ivan Ilych was sending him away he even said straight out: "We shall all of us die, so why should I grudge a little trouble?"
The culture of 19th century czarist Russia was heavily bureaucratic. To be in society was to be a cog in a machine, to do the same work--day in and day out--with little hope for advancement or reprieve from paperwork and frustration. As such, Ivan treated his co-workers with detachment and formality. Private conversations were prohibited. The society was engineered for efficiency; close relationships were not productive. This obviously carried over into his marriage and family life, for he took all for granted.
Even his wife and family are acquaintances only. His wife and relatives and doctor go through the motions of friendship and nursing, but they are not friends or nurses. They are detached, as he is, even in the rituals of sickness and death. Ivan is merely something to be tolerated, and they treat him as a kind of sickness, as a dying man, not as a friend, or even a person.
They do not acknowledge him, as Gerasim does, with dignity and respect. They do not cater to him even when he is ashamed of his condition and does not want to be seen. Only Gerasim exhibits a kind of primitive Christian charity and unconditional love. Ironically, Gerasim has not been conditioned by the bureaucracies; as servant of the peasant class, Gerasim has learned to be humble and long-suffering, kind to the sick. Only he sees Ivan as a human being, not a cog in a machine or a sickness unto death.