Ask Mr. Smarty Plants

Three members of the Salix genus, or willows, are known to be native to Georgia- Salix caroliniana (coastal plain willow), Salix sericea (silky willow) and Salix nigra (black willow). However, it frequently happens that when we are asked about willows, it turns out they were weeping willows, which are non-native to North America. Salix babylonica (Floridata) is native to western China, although it has been used as an ornamental tree for many years in the United States. At the Lady Bird Johnson Wildflower Center we are committed to the use and propagation of plants native not only to North America but to the area in which they are being grown. Because native plants are already adapted to the conditions of rainfall, soil type and climate, they will need less water, fertilizer and maintenance. 

However, in this case, it really doesn't matter if your willow is native or not; your question had to do with the conditions underneath the tree.  Roots of all members of the genus are aggressive and will spread about 3 times the distance from the trunk to the edge of the canopy. They are messy, constantly dropping leaves and twigs, and are prone to many diseases which can leave them with dead areas and an unattractive shape. Groundcovers and grasses simply can't compete with those surface roots taking up all the nutrients, the heavy shade in summer, and the litter that has fallen from the tree. The willow roots also can damage sewer lines and lift sidewalks and paving, and are really suitable only to large properties, where they can be best seen from a distance. It would appear that you will need to accept the bare ground beneath the willow as you would most likely be wasting your time trying to plant more grass.

Salix caroliniana

Salix nigra