My ABS light keeps coming on while driving

Why would you doubt the obvious definitive test? Generator failures on '87 and newer GM vehicles is extremely common. Very often people go through four to six of them in the life of the vehicle. You already did what is needed to prevent repeat failures. That is to replace the battery unless it is less than about two years old. These generators develop huge harmful voltage spikes and the battery is responsible for dampening and absorbing them. It loses its ability to do that as it ages.

Have the generator replaced. The ABS warning light came on because of the low system voltage caused by the inoperative generator. It knows it can't do its job with low voltage.