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The Earth's gravity is much weaker at the moon than at a low earth orbit satellite.The gravity gets weaker  there is no cutoff distance. Theoretically  gravity extends forever. If an object going towards the sun, eventually the sun's gravity would be stronger than the Earth's, and then it wouldn't fall back to Earth.It would happen if  about .1% of the distance to the sun, or about 250,000 km, or 40 Earth radii. This is actually less than the distance to the moon, but the moon doesn't fall into the Sun because it's orbiting the sun, just like the Earth itself is.

So the moon falls toward Earth due to gravity, but doesn't get any closer to Earth because its motion is an orbit, and the dynamics of the orbit are determined by the strength of gravity at that distance and by Newton's laws of motion.