Answer
Your friend is mistaken, for by that reasoning, if I drop a rock here on Earth, it would be sucked away into space to Jupiter's surface, or to the sun's surface, or to a black hole somewhere.
The rock weighs less on the moon's surface (due to the moon pulling it) than it weighs on the Earth's surface (due to the Earth pulling it). This is what we mean by Earth's gravitational field being stronger. But that doesn't mean the Earth pulls on everything, everywhere with a greater gravitational force.
In fact, the moon pulls harder on a rock at its surface than the distant Earth pulls on that exact same rock, still located on the moon's surface.
Rocks dropped on the moon fall toward the moon’s center.
