No, the screen image of a transparency with a density range of 4.0 will not have a 13⅓ stop brightness range, but not because of the inverse square law. The difference between the density range of the transparency and the brightness range of the image on screen will be predominantly because of the factors that AgX has already mentioned: flare, room reflections and some small amount of ambient light being the typical main sources, interacting with the behaviour of the screen itself.
A practical example of this is seen in simple film to video transfer: one cannot reduce the brightness range of the screen image to a range that can be handled by a video camera by simply moving the screen further away from the film projector.
I'll stick with the idea that the main reason for the difference in density range between a B&W negative intended for enlargement onto traditional B&W enlarging paper, and a transparency intended for viewing by projection is not because of the difference in projection distance, but because enlarging paper is matched to a much lower brightness range than that which humans percieve as normal. We don't expect a wide brightness range for all subjects, but the materials need to be capable of it to make something like a sunlit landscape look normal. Even on a light table, a transparency of a sunlit landscape would look quite dull if the transparency had a density range of 1.0, for example.
Best,
Helen