This is a fairly common with Common Grackles, which is most likely the bird described based on color, nest location, and behavior. Most nestling songbirds produce droppings that are in a fecal sac, which the parent birds remove from the nest. Removal leaves would-be predators with no odor from the build-up of droppings that might indicate nest location. Grackles and some other birds tend to drop the fecal sacs in water, probably as an adaptation to further removal of odor from around nesting areas. In urban areas, the water is sometimes a swimming pool, bird bath, or even concrete patios or similar locations, probably because they appear, to grackles, to be water.
So the problem of droppings (fecal sacs) in the birdbath will persist while there are young in the nest (about 10-17 days). Probably the best action is to move the bird bath or cover it during the nestling time, or just clean it daily with the realization that the droppings will cease when the nesting period ends. Moving it so it is less visible or handy to the grackles may stop the problem and yet retain the bird bath for other uses.
Taking action against the nest is difficult, especially in urban areas. In most states, you would need a permit or authorization from the state wildlife agency, even though the Migratory Bird Treaty Act has certain exceptions for grackles.