Question
Is there any way to stop the tree from dropping sweet gum balls?

Answer
There is a new product Florel?brand Growth Regulator (Liquidambar/Sweet gum).
"Florel?brand Growth Regulator breaks down into ethylene, a naturally occurring compound, in the plant. When a plant is under stress it produces ethylene. If the plant is in bloom at the time of excess ethylene production, the first symptom is blossom abortion. This is how Florel?brand Growth Regulator works. It causes the plant to abort its blossoms."

You might ask your local nursery if they can get it for you.
Here is a web link to information on this product and where to get it.
http://www.montereylawngarden.com/faqs/florel.html