How we cite our quotes:
Quote #1
WALTER: I wish you hadn't done that, Hildy... Divorce me. Makes a fellow lose all faith in himself... Almost gives him a feeling he wasn't wanted.
HILDY: Oh, now look, junior, that's what divorces are for.
Walter presents divorce not as sad because he lost the love of his life, but as a blow to his self-esteem. In His Girl Friday, love is presented as a struggle—and especially as a struggle, on Walter's part, for manhood. Can the super-smooth alpha male keep his position as super-smooth alpha male, or will he end up losing out to Bruce Baldwin, of all people? The film is as much about Walter's effort to be a man as about Hildy's to be a woman.
Quote #2
WALTER: You can marry all you want to, Hildy, but you can't quit the newspaper business.
HILDY: Oh! Why not?
WALTER: I know you, Hildy. I know what quitting would mean to you.
HILDY: And what would it mean?
WALTER: It would kill ya.
HILDY: You can't sell me that, Walter Burns.
WALTER: Who says I can't? You're a newspaperman.
HILDY: That's why I'm quitting. I want to go someplace where I can be a woman.
WALTER: You mean be a traitor.
HILDY: A traitor? A traitor to what?
WALTER: A traitor to journalism. You're a journalist, Hildy.
Being a newspaperman (not a newspaperwoman, note) is contrasted with being a woman. Both Walter and Hildy seem to think there's a disconnect between being a woman and having a career. This is perhaps the root of Hildy's misery; if she, and those around her, could admit that women can have careers, she woudn't be so torn up about it.
Quote #3
HILDY: He doesn't treat me like an errand boy, but like a woman.
WALTER: How did I treat you? Like a water buffalo?
Walter treats Hildy often as a subordinate or errand boy, rather than as a wife or a lover. It doesn't occur to Walter that he could treat her as an equal; the best he can come up with, instead, is to treat her as a water buffalo.