God can't prove his existence
It's impossible for God to prove his existence to beings with freedom of thought. For any conceivable thing that God could do to demonstrate his existence, one or more of the following objections can always be made:
Since many of the above objections can't be absolutely disproven, God's existence can't be absolutely proven. This holds even for such popular suggestions as God's writing a message in the sky or God implanting knowledge of his existence in every human (actually, God has already done things similar to both of these). A message in the sky could have come from aliens; an inborn knowledge of God could have been a lie implanted by the alien race that created us, or it could be an evolutionary artifact dating to the time when human-level consciousness first emerged. God has granted us freedom of thought, which includes the freedom to deny his existence. (Incidentally, Jesus pointed out this phenomenon in Luke 16:19-31, and we see it in practice in passages like Mark 3:22 and Exodus 32:1.)
God does provide evidence
Though God hasn't given us proof, he has placed enough evidence in the world for people to believe in him (Rom 1:18-20). Romans 1:20 states, "Since the creation of the world God's invisible qualities...have been clearly seen, being understood from what has been made." How is this so? People throughout history have seen both the power and beauty of nature. Questions such as "What controls thunder and lightning?" and "How did the stars get in the sky?" lead naturally to "Is there someone who created the universe and controls it?" And once someone asks that question, they need only ask God to reveal himself to be given what further evidence they may need to believe (Mt 7:7-8). (For more on this, see How can faith be rational?)
If the evidence is clear, why doesn't everyone believe?
Some have asked, "If the evidence is so plain to see, why doesn't everyone believe in God?" God's existence and Christianity are not propositions like "matter is composed of atoms" that can be regarded neutrally; their truth or falsehood affects our lives and our entire world view. Consciously or unconsciously, everyone has a bias against God. No one wants to stand before a perfectly good God and realize how far one has fallen short of the standard. Thus there is a part in all of us, both Christian and non-Christian, that wants to reject God, just as there is a part in all of us that wants to do good. That bias causes people to act in a variety of ways: some people don't want to investigate any evidence for God, others will try to explain it away, etc.
An interesting thought exercise is to consider how you would react if the evidence you would like God to provide for his existence were presented to you as evidence that an invisible pink unicorn exists. No one wants to publicly vouch for the existence of an invisible unicorn because it would make one look foolish; hence we have an inherent bias against believing in its existence. If we all saw "Hello from the Invisible Pink Unicorns of Planet Qumleb! We wrote this message to notify you of our existence and wonderful powers. Come visit us when you discover high-speed space travel" written in the sky tomorrow, would we take it at face value, or would we think that someone at NASA had pulled off a masterful practical joke (and ridicule those who took the message seriously)?
God wants a relationship built on trust, not proof
Yet even if God provided proof that was satisfactory to everyone, faith and trust would still be required to follow God. The atheist's question would merely change from "Why doesn't God prove his existence?" to "Why doesn't God explain why he did this and not that?" Atheists themselves would become theists, but not all of them would become Christians: one can believe God exists without believing he's worthy of worship, or that Christ's death atoned for our sins.
God wants us to trust him, not just believe he exists. If our every demand for proof and explanation were satisfied, we'd only trust and follow God to the extent that he proved himself to us. We would be relying on the external evidence and our own judgment of it, not actually trusting God. For us to actively trust God, we have to continue in our belief even when what we believe in isn't proven. And why is trust so important? Because it requires a deeper relationship with the one trusted. Anyone will believe a stranger's statement if he immediately produces proof to back it up, but believing a person without having proof requires the believer to have a positive opinion of the person (at least that they deserve the benefit of the doubt) and to take a certain amount of risk. If the risk is large, the believer is dependent on the person. If the trust turns out to be justified, the believer has a higher opinion of the person and a stronger relationship with them.
Another way of seeing it is this: Suppose a married man attends a weeklong, out-of-state conference. His wife can choose to trust that he won't have an affair while at the conference, or she can demand proof of his faithfulness by insisting he call her every hour and give a detailed account of his doings, making him wear a beeper so that she can call him at random, hiring someone to spy on him, etc. Yet if she demands proof, the husband will most likely respond with "What, don't you trust me?" He will be offended because his wife's asking for proof indicates that she doesn't trust him - and since she knows his character, her distrust says that his character is lacking. On the other hand, if she trusts him, it says that she really believes he will be faithful - much more so than if she simply stated she trusted him, yet never spent a night apart from him. It's much the same with God. God wants us to trust him, because that requires both believing that he is trustworthy and acting on that belief. That's not to say that God requires us to have blind faith - he wants us to love him with all of our mind (Lk 10:27), and he will give us reason to believe if we ask him - but that God wants our relationship with him to be built on trust as much as possible (Jn 20:29).
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