Being a physicist, this is close but there is a subtle flaw.... Gravity is pulling on the arrow in both cases. The difference is not in the help from gravity. You have to think of the arrow flight in terms of two orthogonal vectors; one horizontal and one vertical. The difference is gravity only acts on the horizontal vector. If you were to shoot an arrow straight down, gravity is still there but it does not change the impact point. If you shoot an arrow straight across, gravity does change the impact point. This is the reason you base you aim off of horizontal distance.
Yes I am a big nerd