Damage to the moon like that would cause structural integrity issues, eventually leading to the disintegration of the moon into pieces. Said pieces would continue to orbit the planet, but would also run into each other, breaking into smaller pieces. Upon becoming small enough, they would rain down onto the planet as meteors. This would continue until there were no pieces of the moon left in orbit.
This would take place over the course of around 5000 years, during which all life on the planet would be scourged by the equivalent of near constant nuclear fire. The planet afterwards would have extreme seasons and generally be significantly less habitable due to losing the moon's effects on weather, climate, tides, etc.