To attempt to seriously answer the question, what would happen if I declared myself president and refused to step down? Nothing would happen, everyone would ignore me. I’m not the president, and no amount of shouting on my part would make people treat me as the president.
A similar thing would happen if Bush refused to step down. He’d be taken down. His term would expire and a new person would be president. If the old president refused to leave the white house, the new president would order the Secret Service to remove him. And they would. The old president would likely face criminal charges related to his coup attempt. The Justice Department would handle that, I suppose. If the old president has some armed goons and barricades himself in the white house, I suppose the real president would order the marines in. Or more likely an FBI SWAT team, it would look better if the ousting forces were civilian police rather than military.
Thing is, the president is only president because we all agree he’s president. Bush can give all the orders he wants, that doesn’t matter if people don’t obey him. Bush would only be able to declare himself dictator if a majority of people wanted him to be dictator, or at least a very large and powerful minority. And even if a large portion of the military supported the coup, we’d still have civil war as most of the state governments would not support him, most of the federal employees would not support him. And anyway, how many people in the military today would support a coup? Just about zero. How many Republican Senators and Representatives and Governors would support a coup? Just about zero. How many career civil servants would support a coup? Just about zero. How many paramilitary political/thug groups (equivalent to the Brownshirts of the 1930s) would support a coup? Just about zero. How many in the general population would support a coup? Just about zero.
Since nobody would support a coup, what good would it do for any President to stage a coup?