One major moral objection to the existence of God is this: God cannot be both omnipotent (all-powerful) and omnibenevolent (perfectly good).
If God is perfectly good, he would not want any evil, death, or suffering to be in the world. If God is perfectly powerful, he could take all evil, death, and suffering away. As it happens, evil, death, and suffering all exist in this world. Thus, the only options are that God is not good (he allows suffering), God is not powerful (he cannot stop the suffering), or God does not exist (there is no escape).
This kind of contradiction is not uncommon when discussing the possibility of God's existence based on his character. Another contradiction is that most monotheistic religions - Judaism, Christianity, and Islam being the big three - believe that God is both perfectly just and perfectly merciful.
But how is that possible? If God is perfectly just, he must punish every single sin for what it is. If God is perfectly merciful, he must prefer to forgive every person of his sin. So how can you have it both ways? You can't! These things are contradictory. So the God of these monotheisms is logically impossible and thus, must not exist.
It turns out that there is only one monotheism in the world that can answer both these problems. But we'll get to that in a moment.
Let's start with the omnipotence/omnibenevolence problem. If God created the world exactly as we know it today, then certainly, he must not like us very much. If God created the world with cancer, Alzheimer's, a human capacity for torture and other evils, and death, how can he possibly be both good and powerful enough to stop all evil from happening? If God created the world as we know it, and is all-powerful, then he is not good.
It turns out that some monotheisms don't have this problem because they teach the doctrine of original sin. When God created the world, he created it perfectly - free from all the aforementioned evils. It was not God's doing, but Man's doing, which brought evil into the world. Man did this when he sinned against God. When Man sinned, the world became a fallen place, and death entered the scene.
Christianity and Judaism teach the doctrine of original sin. These monotheisms allow God to be perfectly good and perfectly powerful, even with the evil we see in the world today, because they believe that the evil is a result of Man's sin, not God's original creation. They also believe that God will one day return and restore the world to its original state of perfection. Islam, however, does not embrace the original sin doctrine. It teaches that God created the world exactly as we see it today, evil and all. God created Man with his current capacity for evil. God created the world to include death and disease. And yet somehow, God is . . . good? This is a major problem for an Islamic theory of ethics.*
The next problem is God's perfect justice and perfect mercy. It is impossible for God to punish every sin that every person has ever committed, and for God to be perfectly merciful and forgive every sin that every person has ever committed. Impossible, that is, unless God is triune.
Christianity and Christianity alone teaches that Jesus is God, and that he came to bear the punishment for the sins of the world. What does this do for the Christian God? It means that God's justice, his wrath over sin, can be perfectly poured out on the cross. God's perfect justice is satisfied with the death of Christ. And because of Jesus' sacrifice for you and me, God's mercy can be fulfilled as well. If we accept Jesus' death and resurrection, our sins have been paid for, and God in his perfect mercy can delight in forgiving us.
Of course, other monotheisms do not accept Christ's deity. They do not accept his sacrifice. So they are left with a God who is either perfectly merciful (and lets sinners get away with committing horrible deeds), perfectly just (and sends everyone to Hell), or both (which is just a weird contradiction). Only Christianity, with its worship of Christ, can have a rational God who is both just and merciful.
And so these two logical and moral problems with monotheism discredit all monotheisms except Christianity, which stands true with a God who is both moral and rational.