There is a certain irony in this use of the phrase "nearest acquaintances," as actually, the news of the death of their colleague does not produce grief, as we might expect it to, but rather much excitement about who will receive Ivan Ilyich's old position. Note the reaction of his former colleagues to the news of the death of one of their number:
So on receiving the news of Ivan Ilyich's death the first thought of each of the gentlemen in that private room was of the changes and promotions it might occasion among themselves or their acquaintances.
Their overwhelming thought, apart from the possibility of securing a promotion, is the thankfulness of feeling that "it is he who is dead and not I" and then the pain of having to fulfill the "tiresome demands of propriety" by attending Ivan Ilyich's funeral. At one stage the narrator calls these gentlemen the "so-called" friends of Ivan Ilyich, and we can see their insincerity and the way that they, like Ivan Ilyich's family and almost everyone else in this excellent novella, except for Gerasim, ignore the reality and enormity of death and its impact on their own lives.