No, God is not logically required. The "universality" that is greater than our personal experience, and can be drawn upon to formulate a system of ethics, is our collective experience. There is evidence to suggest that a sense of fairness is inherent in primates - capuchin monkeys can be trained to trade pebbles for cucumbers, but if one of them gets a grape (which they like better) the rest will stop trading. Sound like a strike to you? Does to me. It has also been observed that if you rig up a rope that requires two monkeys to pull it to make food drop down, when one of the monkeys refuses to share the other one won't help him pull the rope again - primates will give up a chance at food rather than tolerate unfairness.
We're not capuchin monkeys, but we are primates. Observations of the behavior of our distant cousins in nature - in cooperation with our ability to reason, empathise, learn about the natural world, love one another and pass on our collective history from one generation to the next - provide ample information to construct a system of ethics that is far more consistent with the principle of universality than the system of morals offered by any individual religion or deity.
When there are so many gods and religions to choose from, is there any potential for a universal code of ethics inherent in God-belief? I think not. In fact, I think placing the responsibility for constructing a system of ethics outside yourself lays you open to somebody else (not God) constructing it for you to advance their own agenda.