Why won't my water heater stop filling?
If water is flowing in the white tube, that's coming from the "Thermal Expansion Control Valve" built into the shutoff.  Worst case, this could be because the thermostat or gas control valve has failed on the heater, causing the tank to get too hot.
Is the water too hot?  Is there a Temperature and Pressure relief valve on the tank and is it also tripping? 

If the water heater is running too hot, fix whatever is causing that.  But that Thermal Expansion Control Valve (green handle) may have just failed, either way, it will probably have to be replaced.

Also, it looks like there might be a check valve (back-flow prevention) in that stack.  And/or there may be a check valve on your water supply.  Either way, this can cause unacceptable pressures to build up, tripping the expansion control valve and wasting water and energy.

To solve that issue, install a Water Heater Expansion Tank to keep the system at a safe pressure without tripping any relief valves.