Air does fall under gravity, as other comments. The other side of the coin, the reason that there is not just a few feet of super dense oxygen and nitrogen coating the surface with the vacuum of space starting just above, is due to thermal motion. At surface temperatures air molecules are bouncing around at average speeds of several hundred mph. Without gravity they would fly off into space. So the two factors balance out with the atmosphere stretching out, getting thinner and thinner, to an arbitrary limit usually taken to be 100km above sea level.
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