He is not the King of the US.
You're right, but he's about as close to a king as we'll ever get. The head of state is supposed to extend diplomatic courtesies, be cognizant of the cultural norms and social mores of other nations and abide by them without violating his own nation's mores, and convey the general goodwill of his nation. In a combined head position like the US Presidency, it becomes difficult to separate the dual role as head of government AND head of state, making the 'correct' action in any situation much harder to define. While I think it was incorrect from a diplomatic standpoint for him to bow as low as he did, I think that a smaller one (less AOB, if you will) would have been merited as a sign of respect from one head of state to another, not demonstrating subservience or deferral as is being suggested. I don't think the President was demonstrating anything but respect for the Emperor, as he should as head of state.
However, had he been acting in his role as head of government, I believe the President should not have bowed, since he is one government figure meeting another government figure. However, Akihito is NOT Japan's head of government, Hatoyama Yukio is. Were Obama meeting Hatoyama, I would be much more offended, because THAT would demonstrate political deference and subservience, whereas bowing to a head of state (by definition a cultural and diplomatic representative) is at its core a respectful show of understanding of the cultural differences. Since Obama was acting as head of state on this occasion, meeting another head of state, in my opinion the bow was probably warranted, though not to the degree (both figuratively and literally) to which it was performed. In my eyes, right intentions, wrong execution...right in line with everything else he's done since he took office. (See? You CAN criticize the President and still be respectful IAW Article 88! It's easy!
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This hypernationalistic 'The US bows to no one' stuff comes off as arrogant to me, and in my mind continues to foster the notion that Americans do not care about other nations' social and cultural gestures. I, for one, would like to see that opinion changed.
And for the record, he's not the first president to do so.