Why can't the earth discharge my DC batteries?
If the battery negative was also connected to ground, then it would, as it's completing a circuit. 

The neutral side of house AC power is connected to ground at some point, either the generation station, the sub station, or the inlet to your house, you just don't see the connection normally. It's there so that if there's a live to chassis fault in your equipment, which would be hazardous, enough current will flow to blow a fuse or trip a breaker.