Aciman was born in Alexandria, Egypt, the son of Regine and Henri N. Aciman, who owned a knitting factory. His mother was deaf. Aciman was raised in a French-speaking home where family members also spoke Italian, Greek, Ladino, and Arabic.
His parents were Sephardic Jews, of Turkish and Italian origin, from families that had settled in Alexandria in 1905. As members of one of the Mutamassirun ("foreign") communities, his family members were unable to become Egyptian citizens. As a child, Aciman mistakenly believed that he was a French citizen. He attended British schools in Egypt. His family was spared from the 1956–57 exodus and expulsions from Egypt. However, increased tensions with Israel under President Gamal Abdel Nasser put Jews in a precarious position and his family left Egypt nine years later in 1965.
After his father purchased Italian citizenship for the family, Aciman moved with his mother and brother as refugees to Rome while his father moved to Paris. They moved to New York City in 1968. He earned a B.A. in English and Comparative Literature from Lehman College in 1973, and an M.A. and Ph.D. in Comparative Literature from Harvard University in 1988.