Samsung S4 Screen Problem
My guess is that the underlying hardware that makes the screen run is damaged. In a touchscreen, or all monitors in general, a complex series of wires sends electrical impulses to the individual led diodes (a set of Red, Green, Blue is called a pixel). If one of the main wires became damaged or loosened, it will seriously mess up the screen's viewing output.

Judging by the color difference in your photo, the only part receiving signals from the graphics unit is above a defined line (is that a crack? If so, that is most likely where a breaking event could have occurred).

Since it fell down hard enough to eject the battery, the internal components most likely suffered some damage. If you have a service provider, they might be able to look at your phone in more detail and tell you exactly where the issue is and how (if possible) to fix it.

Best of luck!