No, stuff on the moon does not float. The moon has gravity, just like Earth, and this gravity pulls objects towards its center. This is what keeps the moon orbiting around Earth.

The difference between the moon and Earth is that the gravity on the moon is much weaker than the gravity on Earth. As a result, objects only weigh about one-sixth of what they would weigh on Earth. This makes it seem as though things are floating on the moon, but they are not. They are still being pulled down by the moon's gravity.