Do I need to do anything after the stump is ground out?

Unless the tree was an invasive species CONGRATULATIONS you are done!

HOWEVER, if you did have an invasive tree (fruit-bearing trees, honey locusts, ailanthus, sweet gums, mulberries, hybrid poplars, bamboo and other invasive species), the roots can survive and continue to grow. In these cases saplings will continue to sprout and will have to be cut or mowed over until the plant stops producing them.