Does Falling Comet deal 6 damage at a battlefield if I play a spell reaction to move my unit out of that battle?

No — if you move the unit out of the battlefield in response, Falling Comet will not deal damage to it.

Falling Comet resolves on the Chain last, and at that point your unit is no longer a legal target ("a unit at a battlefield"). The spell still resolves, but the damage instruction is ignored — it doesn't fizzle, and no new target can be chosen (Rule 359.3.e.5, FAQ #6475). Any other instructions on the spell (there are none here) would still happen.