The pilot stated that he departed on a visual flight rules cross-country flight at approximately 0300 local time. Instrument meteorological conditions prevailed at his destination airport where the pilot requested and received an instrument flight rules clearance for an instrument approach. The pilot conducted two missed approaches but fog precluded him from making visual contact with the runway before diverting to another airport. The pilot conducted an instrument approach at the alternate airport and, just prior to landing at approximately 0700 local time, realized that he had failed to extend the landing gear. The airplane subsequently landed gear-up, resulting in substantial damage to the fuselage. The pilot stated that fatigue, weather conditions, and the distraction of the diversion contributed to the error.