Thank you ..!
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Look no further than the U.S. Constitution, which states in Article I, Section 9, that ‘No title of nobility shall be granted by the United States.’ Those weren’t just words that prohibited Congress from naming someone a prince or princess, duke or duchess, lord or lady.
Those words were clear signals that in the U.S. all persons are on equal footing: that no nobility would exist here and thus no one had to bow to anyone.
Certainly people here have titles such as president, chief executive officer, mayor, chancellor, and the like, but none of those titles was encoded on someone’s DNA. Titles were to be ascending, earned through one’s own sweat equity and remarkable character, rather than descending, simply a generational bequeath to one’s progeny.”
So a US citizen when meeting a king or queen – in the United States or in the monarch’s country — should simply offer nod of the head as a sign of respect and shake the hand of the monarch if it was offered. This contrasts with either a deep bow or curtsy which would be an appropriate sign of fealty from a subject.
Regarding the President. again I would quote of Chris Young, when he says both are “equals on the world stage. Both are heads of state …. the only order of precedence that exists between the two is usually an alphabetical one rather than one of rank.”
Since they are peers neither would bow to the other. So no, the President of the United States would perhaps offer nod of the head as a sign of respect and shake hands.