America was founded upon a Constitutional Republic that sought to get away from the divine rule of monarchies. It is a fundamental component of the Age of Enlightenment and Classical Liberalism.
America was also founded upon that fact that US citizens can become the President without having to take a DNA test to prove your royal hereditary. I wish more President respected this fact and held their head high when they meet with foreign dignitaries who are dictators or represent hereditary monarchs.
When you bow to hereditary monarchs and dictators, you are forgetting some of the fundamentals principles that this country was founded upon and placing politics before principles.
As for Japan, I can tolerate a bow (doesn't mean I like it) and I am not going to dig up the past with them since I view them as an ally nowadays. As for bowing to Abdullah, I think it is a disgrace, especially given his civil rights record, but I will take a bow over holding hands and kissing him.
So, before Republicans go pointing figures at Obama bowing to foreign dignitaries, there are three fingers pointing back at you.