The Ancient Greeks have no verifiable evidence of gravity, and thus should conclude gravity does not exist. Of course this is a false conclusion.

Absence of verifiable evidence is insufficient to conclude something is false. You'd need to be able to show its negation is verifiable to show anything about the truth of the existence of God.

It is perfectly fine to assume God does not exist for the purpose of doing science, and that's a perfectly rational position to take. However, it is not a proof, so you have not done enough to show the public that god does not exist.