Look, bowing the to the Saudi king was not a good idea.
But bowing to the Japanese Emperor has the cultural equivalence of shaking the man's the hand. While in some cultures it may be a sign of subservience, to the Japanese it's just common courtesy.
I think this explains it pretty well, even allowing people to still heap some criticism on Obama simply for the awkwardness in execution.
And for those people complaining that our president shouldn't show common decency to the Japanese emperor because of the history of WWII, need I remind you that we've had a pretty darned good relationship with England, and its leaders, for quite some time now (even if we may bumble the gifts we give them from time to time).
There's more than enough to criticize Obama for. Let's not delegitimize those things with this sort of silliness.