This question’s answer must begin with a categorical study of the two concepts. To briefly examine them for the sake of time, atheism is a belief that denies the existence of God, and thus limits intelligence to mankind exclusively. Morality concerns right and wrong, concepts that are extremely vague. As a matter of fact, the terms right and wrong beg definition, or perhaps it is better to say that they are subject headings in the essay of one’s worldview.
One’s belief system dictates what is right and what is wrong, and thus all worldviews contain morality concepts that are defined by that worldview. It can then be said that atheism and morality are not incompatible, because morality is not inherently incompatible with anything.
But any worldview that makes a statement about God restricts what it can possibly say about morality.
So what must an atheist believe in the realm of ethics and why?
The most important ethical implication of atheism is the lack of objective ethics, that is, ethics that apply to everyone. If there is no God, it follows that the only intelligence in this world is that of humans. Every individual finds themselves cast into this world alone, required to choose responsibly, where the responsibility for choices made, falls on you.
Why Atheism and Morality are Compatible
Nietzsche, perhaps the most honest and insightful atheist this world has ever known, stated in the thirteenth section of his work entitled The Anti-Christ that, “Let us not under-estimate this fact: that we ourselves, we free spirits, are already a ‘transvaluation of all values,’ a visualized declaration of war and victory against all the old concepts of ‘true’ and ‘not true.'”
The problem that one runs into in society is that morality is often misused to mean objective morality, and in this sense atheism is not compatible with morality. Atheists cannot criticize another on an ethical decision honestly; they have no foundation for it.
The have been arguments, sometimes moderately convincing, for nature-based ethics; what is destructive to life is bad, what is productive to life is good. This is an interesting observation if the conscious destruction of life seems bad, but there is no philosophical basis for this belief. The question remains, why is the destruction of life bad? Because you feel like it should be? It can be said that morality is compatible with atheism, but objective morality most certainly is not.