Just to check, are you asking about a tiny cable on the surface of the Moon, or a 1-light-second-long cable where both ends are “the bottom” in local gravity even though they’re on opposite ends of the straight cable?

Oh, thanks, I see the problem now.

I imagine it would be nailed down at the Moon, even though it 's going "up" on that end too, the Earth end will very likely weight much more than the Moon and pull the entire cable. But the fact that it must be much fatter near the Moon does indeed complicate things.