Yes, An American can marry into the British Royal Family, although Roman Catholics are barred from ever inheriting the throne.
King Edward VIII abdicated the throne not because he wanted to marry an American, but because she was an Divourcee.
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Americans have married into the royal family as close as about #40 in line for the throne.
King Edward VIII was unacceptable as a king for a multitude of reasons. At many times in his life he told people he did not want the responsibility. He had a love life that made many people cringe, since he was almost always involved with bossy married women. The final relationship that resulted in marriage was to a twice divorced American woman. If everything else had been acceptable, the public might have accepted his marriage to an American.
The first visit of a monarch to America was by the brother of Edward VIII (King George VI the father of Queen Elizabeth) in 1939 (3 years after Edward resigned).
Personally, I feel that Edward's personal behavior was so offensive to many British that the monarchy would have been disbanded in WWII (like so many of the continental monarchies).
It should also be noted that no one can ever marry into the British royal family and inherit the throne, one must be born into the line of succession. There are no rules to say that the sovereign's wife or husband cannot be a born foreigner - Wallis Simpson's nationality was not the issue, and indeed the current Prince Consort, Phillip, was born in Greece. That's why there is no fear that a foreigner could become sovereign, the throne can only ever be inherited by a born royal.
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