It does, but as more air drops down, the pressure increases. This pressure then starts to push back against the air above it. Which is why we have atmospheric pressure at the surface, but that goes down to pretty much 0 in space.
Even in low earth orbit there are still some particles, which causes satellites and such to slow down, requiring them to fire some thrusters every once in a while.
Why and how does atmosphere exist? Shouldn't all oxygen (and everthing else) fall down due to gravity?