There are a lot of "safety" procedures in the military that aren't done in the civilian world. Even in the civilian world there are differences. Each airline even has its own safety procedures and even write their own manuals. "Check Gear Down/Gear Down" is just one of them that the military established long ago as a precaution to avoid stupid mistakes that are very avoidable. Call "Gear Down" and actually look at the gear lights to be sure you have 3 green. No one wants to be the one idiot that forgot the gear. As has been mentioned, in two pilot civilian cockpits this is done between the pilots. It gets very busy in the traffic pattern and stupid mistakes happen. "Gear Down", as well as the many other SOPs, keep it all standardized and helps make sure that its always done the same way by everyone, everywhere. In a huge organization like the USAF such things are very helpful when you have personnel constantly training, coming and going and transferring etc.
Similar calls are "call the ball/roger, ball", "feet wet/feet dry", "fence in - check fuel flows are good, fuel levels are good, IFF is stdby, oil pressure good, fuel tanks and oil system pressure off ( in case you take a hit ), PC system all good, weapons selectors set, weapons master off, chaff/flares are armed, etc..." (when crossing the hypothetical "fence" into hostile territory when on a combat mission ). Another neat one is calling "Base Plus" when climbing above Controlled Airspace above 60,000 feet which can still be heard from time to time.