Hypothetical example:
So there you are as POTUS, and you are meeting with the Emperor of Japan in Tokyo to strike a mutually beneficial trade deal, but it will benefit the US more than Japan.
You are told that upon greeting him, if you refuse to bow and show respect for Japanese customs then the Emperor will consider you rude and refuse to do business with your country for the rest of your administration.
So you can swallow your pride and adhere to Japanese customs for a day to get what you want for the country, or you could take the "I bow to no one" stance and walk away with nothing.
This is the way politics works. Sometimes you have to give a little to get a lot more. The hard-charging, "we're too proud to follow your customs" diplomacy would get us nowhere.
The Emperor of Japan has no more political power than the Queen of England. It is NOT a Japanese custom to bow and shake hands at the same time….ever. It is not a Japanese custom for the leader of a foreign country to bow in a submissive manner before the Emperor. If Obama was hypothetically told by his protocol people to do as you state…they need to be fired for being idiots.