The president had this to say about "the bow" to Japanese Emperor Akihito:
"I’m representing the United States of America. And we’re talking about a friend, and we’re talking about an ally. We’re talking about a nation with whom we have constructive relationships."
That wasn't President Barack Obama saying that about his recent bow to the Japanese Emperor. That was President George H.W. Bush saying that in 1989 in response to a reporter's query about his bow to the new Japanese Emperor.
Presidents bow. President Richard Nixon did so when he visited China and bowed to Mao Tse-Tung. President Obama did so when greeting the Saudi King and the same Japanese Emperor. Presidents even hold hands with other leaders, as President George W. Bush did with Saudi King Abdullah (I didn't hear the Veep complaining about that).
Despite what former Vice President Richard Cheney said, it's really not a big deal. Presidents of both parties bow. It's a sign of respect in many cultures. Just because we don't do it in America doesn't mean it signals weakness when an American president bows to a foreign leader, especially an ally. Now, if they had curtsied, that might be another story...