The pilot of a retractable landing gear airplane reported that, while on downwind to land, he did a partial checklist and did not verify the landing gear position. While on final approach, about 20 ft above ground level, he heard over the radio that his gear was not down. The pilot reported that he immediately extended the landing gear, applied full power and applied back pressure on the control yoke. With the airplane still in the landing configuration, he believed that the airplane stalled and subsequently landed hard, separating the nose gear. The airplane veered off the runway and struck a runway sign before it came to rest upright. The left wing was substantially damaged. The pilot reported that there were no preaccident mechanical failures or malfunctions with the airplane that would have precluded normal operation.