Difficult question, and I doubt there is an objective answer. Tom Au's answer is correct, but I believe it is.... perhaps incomplete... "Gentleman" was an important cultural concept for Americans of the period, difficult to define, but very much present in social and political affairs.
I believe Gordon Wood in Revolutionary Americans characterized "Gentleman" as someone who was more independent than a tradesman. He provided the example that Benjamin Franklin sold his business(es) before entering politics. (I am paraphrasing an entire chapter of a book; this is not a direct quote)
Constitutionally, there are no gentle men or women in America. That said, much of the cultural context of the Constitutional convention was arranged around removing political power from tradesmen, and the laboring classes. America rejected the English class based society, but that was a rejection of class from birth, in favor of class based on outcome. (If I recall, this was one of the major themes of McCullogh's biography of Adams).
There was a general recognition that people who were dependent on others for their livelihood were not truly independent. Laborers, and tradesmen (historically there was a third category, but it escapes my mind) were dependent on others for their livelihood. Their opinions must never diverge too far from their customers and patrons.
Large landowners were truly independent. They did not rely on anyone, so they could be free to think deeply and to argue their opinions without fear of reprisal. (Any reading of American history will reveal that reprisal was a major force... Why was Madison not a Senator? Burr vs Hamilton? etc.)
Although America had renounced an inherited aristocracy, they still had a keen appreciation for honor. Hamilton's duel with Burr was over honor. Washington nearly refused to participate in the Constitutional Convention because he had pledged to retire from public life and going back on his word would diminish his honor. Jackson fought duels. Once you start looking, it is easy to develop additional examples. A gentleman had to behave as though he valued his honor and his reputation as a gentleman. (and only someone who was financially self-actualized could actually do that; tradesmen were perceived as unable to challenge those on whom their livelihood depended).
A Gentleman in a European country was an artifact of birth. In the US, Gentle was not a status granted by birth, but by some combination of self-actualization, wealth, education, reputation, and other factors. Lawyers considered themselves gentlemen (esquires) - likewise other prosperous professionals (who could, in theory, ignore the hostile opinions of others.)
I think this interpretation of "gentle" is more in line with the use of the term "gentleman" in Tom Sawyer - the couple of critical reviews I consulted point out that the irony is that almost all the men in Sawyer are definitely not gentlemen and would not have been treated as such by their neighbors.
Bit of an abstraction with more indirect references than I prefer in a good answer, but I think it is a useful complement to Mr. Au's.