In watching tons of old movies on TCM, I have noticed that Americans used to say: "How do you do" when meeting someone. We do not say these words anymore and instead now say: "Nice to meet you." I prefer to say: "Nice to see you," in case I have met this person before and do not immediately remember them.
When being introduced to The Queen, or any Royalty, always wait for the introduction, wait for the Queen to extend her hand and wait to answer her by saying: "How do you do," NOT "Nice to meet you." "How do you do" is more formal and is preferred by Royalty as traditional.
When Prince William's girlfriend, Kate Middleton, met the Queen, she said: "Nice to meet you" and was unmercifully bashed by the media. The press said she insulted the Queen by not saying: "How do you do." The press called her "common" and "rude," and all because she did not know to say: "How do you do" to the Queen.
Sounds like such a stupid little thing, but apparently not. That is what etiquette is all about....knowing all your little rules.
I hope President Obama and his wife have had their etiquette lessons and know all their Royal rules.