Spiritual death is a term used by evangelical Christians. What do we mean however by that term? Morris Ashcraft gives a good definition,.
"With reference to sin, death means much more than that which the coroner certifies. It means separation from life because it is separation from God. Physical death is a separation from life, in one sense; but in the case of Jesus Christ who was raised from the dead, the separation was not ultimate. Biblical faith posits the fearful possibility of an ultimate separation from God and life which is called "the second death" [Morris Ashcraft, Christian faith and Beliefs (Nashville, TN: Broadman Press, 1984) 199.]
Separation from God is appropriate. When Adam and Eve were cast out of the garden, they were separated from Paradise. They could no longer walk with God in the cool of the day. They were under a curse.
The Hebrew word used in Gen. 2:17 is "mooth" which is a primitive root meaning "to die (literally or figuratively)" [James Strong, Strongs Exhaustive Concordance of the Bible (Nashville, TN: Thomas Nelson Publishers,) 63]. The idea is that of a condemned prisoner; someone they schedule to die but is awaiting execution. Adam was a dead man by decree, but was still awaiting the carrying out of the sentence. The apostle Paul expressed it this way, "As for you, you were dead in your transgressions and sins," [Eph. 2:1]. God was giving Adam solemn warning that to disobey would bring cessation of immortality. This is evident in Jewish thought also. Adrien Janis Bledstein wrote "eat of it; for in the day that thou eatest thereof thou shalt surely die," or "be doomed to death""[Adrien Janis Bledstein,"The Genesis of Humans: The Garden of Eden Revisited." Judaism 26, no. 2 (Spring77 1977): 187. Religion and Philosophy Collection, EBSCOhost (accessed January 25, 2011).] Christianity's root is in Judaism. The early Christians would have placed all the teachings of Christ in the framework of Judaism. So Morris and Bledstein agree with the meaning of Genesis 2:17. The Presbyterian commentator Matthew Henry also echoed the thought, "Thou shalt become obnoxious to death, like a condemned malefactor that is dead in the law"" [Matthew Henry, Matthew Henry's Commentary on the Whole Bible Volume 1 Hendrickson Publishers, 1991 14]
So what died that day as God drove Adam and Eve out of the garden? The soul of man died. Man was now separated from God, he has been cast from God's presence and cursed for his disobedience. Physically man died also. Adam did not drop dead, but he was under a death sentence. Adam now had the fearful awaiting of that death. For the remainder of his life he would wonder when death would come and what it would mean. The same is true of all people today. Everyone knows that death is coming and fears what it will bring. The Apostle Paul understood that Adam's death was a curse to all people when he wrote "as in Adam all die" (1 Cor. 15:22). How God transfers sin and death through the generations is much debated. There is no doubt from everyday observation however that it is true.
This brings to the forefront the question what would have happened had Adam and Even not sinned? One can only speculate. It is clear that as long as Adam remained in obedience he would have preserved his immortality. Then what if an offspring had disobeyed? Would Adam have remained in paradise and immortal while his offspring was cast out? What if Eve only had disobeyed? Would she alone have been cast out? These are irrelevant questions because Adam and Eve both ate of the fruit and brought condemnation. Sin has entered the world and so death by sin. The whole matter is summed up in Romans 10:13, "For all have sinned and fall short of the glory of God"