No. Jesus’ death was because of our sin. Therefore, had sin never entered the world, Jesus would not have had to die. This question, however, can only be answered in a “theoretical” way since Adam, Eve, and all of us did sin. Jesus
Though this question is hard to answer in a short and simplified way, let’s consider an analogy. Let’s say that your parents ate poison. The result of this poison is death. The only cure for this poison is to receive a transfusion of new and healthy blood from someone not affected. By analogy, you could say that Jesus entered the world without any effects of this “poison” so that He could offer the divine “transfusion” to Adam and Eve and all their descendants affected by the poison of sin. Thus, the blood of Jesus is what heals us as we receive this blood poured forth from the Sacrifice of the Cross. We receive His saving blood by accepting Him into our lives, particularly through the Sacraments and faith.
But this question brings up another more interesting question. If Adam and Eve (and all us who descended from them) never sinned, would God the Son have become human? Would He have taken on human flesh through the Incarnation by the Virgin Mary?
Though the death of Jesus was on account of our sin, His Incarnation (becoming human) was not solely so that He could die for our sin. The Catechism of the Catholic Church describes that one of the primary reasons for His Incarnation was “in order to save us by reconciling us with God,” but it also recognizes three other reasons: “so that thus we might know God’s love” “to be our model of holiness”; and “to make us ‘partakers of the divine nature’” (See CCC #457-460).
Therefore, it is speculated by some that even if there was no sin, God would have become flesh so as to accomplish these other effects of the Incarnation. Perhaps that’s a bit deep and is only speculation, but it’s beautiful to reflect upon nonetheless!