To be honest, Jeff Davis and Saint Robert, Patron of the South, are to blame for that. Their plan for the invasion of the North cost the Confederacy thousands of good soldiers at literally no gain, and in the first case of invasion, destroyed their chances of aid from Europe. Then again, the Emancipation Proclamation might have caused enough opposition in the North to Lincoln's policies for Lincoln to lose the election of 1864, assuming that a more competant Southern general or perhaps Lee dispatching a few divisions to Georgia could have kept Sherman from taking Atlanta.
As for the OP, no way in hell. First off, the Army is loyal to the government, and no American forces since the half-hearted attempt at an anti-Roosevelt coup in the 1930s have seriously contemplated being kingmakers. Plus, by the time the American people got seriously pissed enough to even remotely contemplate such an event, Bush had only a few months left anyway.
Finally, given that Bush managed to maintain pretty high approval ratings from 9/11 to well into his second term, I don't see the American people as hating him enough to overthrow him in any event. Katrina may have hurt his approval ratings badly, but let's be honest here: the only note the average American took of it was a spike in gas prices and the news services covering the hurricane damage for a long time.