No, God cannot do everything. God is holy and He cannot sin. The Bible tells us He cannot lie (Hebrews 6:18; Titus 1:2). Also, since God is eternal by nature (Psalm 90:2), He cannot stop being God. He cannot deny Himself (2 Tim. 2:13). God cannot be tempted by evil, nor does He tempt anyone with evil (James 1:13).
Some critics of Christianity will say that if God is so powerful and He can do anything then He should be able to make a rock so big He can’t pick it up. If He could do that, then there would be something He could not do. But such absurd and illogical challenges are meaningless. The truth is that God must be consistent with His own nature, and He cannot violate His own nature. Therefore, God cannot lie, cannot stop being God, cannot deny Himself, and cannot be tempted by evil. Why? Because He is God.
The thought that God could sin violates all we are told about God in the Bible. Unless one reduces Scripture as we have it today to fallible and merely human transmission, without any Divine oversight/influence, I cannot see how any genuine Christian could even think that thought, much less express it publically, as I have actually heard of late.
The Battle Cry 
I’ve been asked that question before about the Rock…by christians and non-christians alike. There is this idea that there are no restrictions on what God can do regardless of what the Bible says about it. He can’t deny Himself, so anything that goes against who and what He is, He can’t do. I’ve heard then, that it’s not that He can’t, rather He won’t. The problem is that Scripture doesn’t say won’t. It says can’t and there’s big difference between those two words.
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I never even though it a question – concerning God’s having the ability to sin, so when a professing Christian told me that ‘divine nature’ can sin, my jaw dropped.
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lol. Can God sin. That’s a good one. Especially since it is Him who defines what sin is. Not us. He specifically says He can’t deny Himself, including the standards He has set.
Now, I suppose if the definition of divinity just means immortal, then yeah. Divinity can sin…we see that with fallen angels. But God isn’t just divine. He is the eternal creator, not just another immortal created being.
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