What adhesive can I use for adhering glass stones to a plastic white birdbath (silicone melts in the sun)?

liquid nail

You have got to be using the wrong type of silicone. Try GE's XST silicone II caulk, stuff is great.

Or you could use PL 400 urethane caulk.

try gorilla glue my family used it on mine and it stady

You were not using pure silicon caulk. It will resist heat and water better than just about anything. You do have to hold the stones in place until the caulk sets up.

if you can I would try a hot glue gun, they are mildly pricey at a craft store an last forever and you can use them on anything and it is almost like super glue for the outside weathering

Cement mix a small amount as needed.