Having had time to reap the controversy because bowed in front of King of Saudi Arabia, President Obama reaped more controversy because of bowing before the Emperor of Japan.
Several parties in the United States considers Obama violated the protocol, because the bows seemed to indicate a lower position than the King of Saudi Arabia or the Emperor of Japan, and as president and head of state, its position is equivalent to a king or emperor. But Obama said the party was merely a form of respect only.
Unlike the U.S., these bows exactly a warm welcome in Japan, and even got praise from various circles in Japan.