I was inspired with this question when a coworker asked how to say "Gesunheit" in Ancient Greek. How would an Ancient Athenian, or someone living in the Roman Empire have responded to a sneeze? Is a perfunctory blessing a (relatively) recent concept? Would we have any records of this to begin with?
The ancient Greeks were very superstitious, and a sneeze could be taken as an omen. There's a passage in Xenophon's Anabasis, a work which describes a 4th century expedition against the Persian empire, in which a sneeze is taken as a sign from Zeus. Xenophon has just made a speech to the army when someone sneezes, and the soldiers all immediately bow in worship, interpreting the sneeze as a sign of divine approval for Xenophon's plan.