Can shooting a window with a Nerf gun damage the window?

No the Nerf ball can not break the window. Nerf balls are generally made from a spongy cellular material produced by the reaction of polyester with a di-isocyanate while carbon dioxide is liberated by the reaction of a carboxyl with the isocyanate. Polyester resin reacts with a compound while CO2 is simultaneously released by another reaction. It is this gas that creates open pockets within the polyurethane that, in turn, makes the material soft and light. These soft balls are not that much solid that it can even break the window. These balls are generally designed not to provide any harm while playing with it.