According to Bowie himself at the time of the album release, "It came from reading an article about Kellogg Brown & Root, a subsidiary of Halliburton, the company that Dick Cheney used to run.
"[1] Biographer Nicholas Pegg wrote his own interpretation of the song: "The key on this occasion is the fearful predicament of global politics at the time of the Reality sessions.
The album was recorded during the preamble to, and the prosecution of, the Iraq War, and it's impossible to hear lyrics like 'I don't care much, I'll win anyway.../I'm goddamn rich, an exploding man/When I talk in the night, there's oil on my hands' without pondering their most obvious resonance.
It wouldn't be particularly extravagant to surmise that the 'Moon' of the title suggests the Crescent Moon of Islam, thereby narrowing down the candidates for 'Fall Dog' fairly decisively.
[1] "Fall Dog Bombs the Moon" is based on short melodic motifs, the kind that seemed to permeate his previous album, Heathen.