Bowing to the head of state is more done out of respect than any religious duty.
And before you get all high and mighty on the monarchists, keep in mind that many people currently feel the same sort of reverence to the current president. Even worse, many feel this towards the latest hollywood starlet.
A friend once told me monarchy is a kind of religion and we should respect them for that alone.
Fuck religion then, as well as the military and the monarchies.