When Tom hits Myrtle in The Great Gatsby, he is revealed to be ignorant, abusive, unenlightened, sexist, egotistical, selfish, and a bully.
He is a patriarch of patriachs, dominating his male world at a whim. He sees women and those who are of a lesser social and economic class than he is as beneath him. He lives in a world that just recently allowed women the vote, and he behaves accordingly. He and his family (although he disrespects Daisy, also, when he is with her) are better than Myrtle and hers, and though he will have an affair with her, he will not tolerate her thinking she is an equal.
In Chapter One of F. Scott Fitzgerald's The Great Gatsby, Nick describes Tom Buchanan's physical accomplishments and his extreme wealth. But, both of these do not seem to be enough for Tom; Nick states that he feels that
Tom would drift on forever seeking a little wistfully for the dramatic turbulence of some irrecoverable football game.
Tom has a propensity for violence as evidenced in his swinging and hitting Mrytle in such a cruel and threatening way. He also has a double standard of behavior: Whatever he and his family do is all right, but no one else can take aim at any one of his class. This attitude is evidenced in his description of the "responsibility" of the white race:
It's up to us who are the dominant race to watch out or these other races will have control of things.
Tom wishes to remain in control in his personal life; to be dominant in every situation is his desire.
I guess the first aspect of Tom Buchanan's personality that is revealed here (in my opinion) is his arrogance. It seems that Tom thinks that he (and even his wife) are so much better than Myrtle Wilson that she should not even mention Daisy's name. That seems really arrogant -- it's like Myrtle's good enough for him to have an affair with, but not good enough to even say his wife's name.
The other thing I think we see here is what a bully Tom is. He is rich and he is strong physically. Here, he uses both of these attributes to abuse someone who is neither rich nor strong.