After blowing up the Moon, making it lose more than half of its mass, what did Koro-sensei do to prevent it from crashing onto the Earth? Given the loss of mass, the Moon would have lost the centrifugal force that kept it from crashing onto the Earth. So, there must be something that he done to prevent it, because if not, the World Leaders would've noticed and said something about the Moon getting closer.
Physically speaking, assuming there weren't any debris from the moon blowing up and hitting the earth, there's no real reason for the moon to fall down.
Gravity is determined by (among other things), the product of the Earth's and the Moon's masses. The acceleration is determined by the force of gravity divided by the moon's mass, so if you do the math, the only thing that matters in this problem space is the Earth's mass. The moon has no say in it.
It's similar to the reason all objects fall at the same acceleration on Earth, regardless of mass/weight.