Question

What evidence do we have that the space between the stars is not completely empty?

Answer 1

Err... It depends on how you define empty. Some may say, "It's empty! it's a vacuum that has no objects with which can interact!" A Physicist would say that it is filled with millions of atoms and photons. Of which, the latter is the correct of course.

So, you ask for evidence, but really you might want to be more specific. Because, there is no true refusal to "Space is empty." Because, it simply isn't empty. It's filled with countless particles.