Touch not the Lord’s anointed
People who support the idea that God should be the one to punish murderers often refer to the record of David and Saul, in which Abishai, one of David’s men, wanted to kill King Saul, but David restrained him.
1 Samuel 26:8-11
(8) Abishai said to David, “Today God has delivered your enemy [Saul] into your hands. Now let me pin him to the ground with one thrust of my spear; I won’t strike him twice.”
(9) But David said to Abishai, “Don’t destroy him! Who can lay a hand on the Lord’s anointed and be guiltless?
(10) As surely as the LORD lives,” he said, “the LORD himself will strike him; either his time will come and he will die, or he will go into battle and perish.
(11) But the LORD forbid that I should lay a hand on the Lord’s anointed. Now get the spear and water jug that are near his head, and let’s go.”
This section of Scripture is not applicable to the death penalty. Saul was an ungodly person, but he had not broken any law for which David could execute him as a criminal. More importantly, David was not representing the civil authorities, who, under God’s laws, had the authority to enforce the death penalty. If David had killed Saul, it could never legitimately be called “enforcing the death penalty.” At best, David would have been acting as a vigilante, which is outside the will of God. While it could be argued that David could have killed Saul and called it self-defense or war, David did not need to kill Saul for either of those causes, and he chose not to.
Although, in both self-defense and war, there are times when killing is essential to survival, it is also true that there are times when there are other options such as taking prisoners or running away. David had no qualms about killing in war, as the Bible amply testifies. The story of David and Goliath is well known, and David fought and killed so many people in establishing his kingdom that God would not let him build the Temple. God said to David:
1 Chronicles 22:8-10
(8) You have shed much blood and have fought many wars. You are not to build a house for my Name, because you have shed much blood on the earth in my sight.
(9) But you will have a son who will be a man of peace and rest, and I will give him rest from all his enemies on every side. His name will be Solomon, and I will grant Israel peace and quiet during his reign.
(10) He is the one who will build a house for my Name.
David not only killed in war, he also protected his kingdom by commanding that troublemakers be put to death, including Joab (1 Kings 2:5-7) and Shimei (1 Kings 2:8,9). A study of David’s life makes it clear that when he spared Saul’s life, it was not because David was against the death penalty or killing in war. Genesis 9:6 is clear that mankind must enforce the death penalty for murder, and the rest of the Bible corroborates this. As our study continues, we will see that nowhere does God ever state that He will take over the responsibility for executing criminals until His final judgment of them. The language all through the Bible is that man will have to do it. 
 The entire Bible is “the truth,” yet there are things in the Bible that are not true. How can this be? When the Bible records a man speaking, it records what he said and therefore is a “true” record of what happened. That does not mean that the words the man spoke are the truth. They may be out-and-out wrong, such as when the religious leaders said of Jesus, “He is possessed by Beelzebub! By the prince of demons he is driving out demons” (Mark 3:22). It may also be the case that a man would speak by revelation to a specific situation, but what he said did not apply in every case. Elisha told Naaman that if he dipped five times in the Jordan River, he would be healed of leprosy (2 Kings 5). Yet it is obvious that not all lepers are cleansed by washing in the Jordan, so while Elisha’s statement was true for Naaman, it is not true (i.e., does not apply) to everyone. Another example of this is when David said of Saul, “the LORD himself shall strike him” (1 Sam. 26:10). This was not a part of the Law of Moses, and David obviously did not believe this to be true for every evildoer, because David executed the man who claimed to kill Saul (2 Sam. 1:15,16) and the men who killed Ishbosheth, Saul’s son (2 Sam. 4:12). David was speaking specifically about Saul, not about how to run the justice system of a government.