If Adam and Eve didn't eat the fruit, would Jesus still die on the cross?

​No. Jesus’ death was because of our sin. 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 and Eve did sin, as have we all.  
 
Though this question is difficult 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 His blood poured forth from the Sacrifice of the Cross. We receive His saving blood by accepting Him into our lives, especially through the Sacraments and faith.  
 
But this question brings up another more interesting question. If Adam and Eve (and all of us who descended from them) never sinned, would God the Son have become human? Would He have taken on human flesh through the Incarnation through 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 explains that one of the primary reasons for His Incarnation was “in order to save us by reconciling us with God.” The Catechism identifies 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).  
 
Some speculate that even if there were no sin, God would have become flesh so as to fulfill these other effects of the Incarnation. Perhaps that’s a bit deep and is only speculation, but it’s beautiful to reflect upon nonetheless!