Depends how much Bob is getting paid. He'll do anything for money.
Should Bobby Fischer rematch a woman?
If by "rematch" you mean "slap around", then totally. Otherwise no, unless she makes him a sandwich first.

If he did win against the female opponent, should he say "good game"? This would be either mandatory or very rude, I forget which.

Quote below was when Fischer was young. Whether he felt that strongly later on in life, who knows. We do know that no woman ever played in tournaments Grandmaster Fischer competed in. None qualified for the Mens Grandmaster title, let alone scaled the heights to be in the Top 100 players in the world.
They're all weak, all women. They're stupid compared to men. They shouldn't play chess, you know. They're like beginners. They lose every single game against a man. There isn't a woman player in the world I can't give knight-odds to and still beat. -- Bobby Fischer
Fischer is Fischer, but a horse is a horse. -- Mikhail Tal (upon hearing Bobby Fischer’s claim that he could beat any female player in the world giving her knight odds)

Quote below was when Fischer was young. Whether he felt that strongly later on in life, who knows. We do know that no woman ever played in tournaments Grandmaster Fischer competed in. None qualified for the Mens Grandmaster title, let alone scaled the heights to be in the Top 100 players in the world.
They're all weak, all women. They're stupid compared to men. They shouldn't play chess, you know. They're like beginners. They lose every single game against a man. There isn't a woman player in the world I can't give knight-odds to and still beat. -- Bobby Fischer
Fischer is Fischer, but a horse is a horse. -- Mikhail Tal (upon hearing Bobby Fischer’s claim that he could beat any female player in the world giving her knight odds)
That quote by Bobby Fischer was around 1961 and was published in the January 1962 issue of Harper's Magazine. Years later in the 1990s he would refuse to face a young Judit Polgar.

In an interview with Ralph Ginzburg published in the January 1962 issue of Harper's Magazine, future World ChampionBobby Fischer was quoted as saying that women were weak chessplayers and that he could successfully give knight odds to any woman in the world.[17][18][19] Fischer later claimed that Ginzburg had distorted what he had said.[20]
https://en.m.wikipedia.org/wiki/Chess_handicap
Quote below was when Fischer was young. Whether he felt that strongly later on in life, who knows. We do know that no woman ever played in tournaments Grandmaster Fischer competed in. None qualified for the Mens Grandmaster title, let alone scaled the heights to be in the Top 100 players in the world.
They're all weak, all women. They're stupid compared to men. They shouldn't play chess, you know. They're like beginners. They lose every single game against a man. There isn't a woman player in the world I can't give knight-odds to and still beat. -- Bobby Fischer
Fischer is Fischer, but a horse is a horse. -- Mikhail Tal (upon hearing Bobby Fischer’s claim that he could beat any female player in the world giving her knight odds)
That quote by Bobby Fischer was around 1961 and was published in the January 1962 issue of Harper's Magazine. Years later in the 1990s he would refuse to face a young Judit Polgar.
he refused to play anyone, not just Polgar
Cathy Forbes in 1992: "Why, in your opinion do women generally not play chess as well as men?"
Fischer: "Well, it seems to be nature, but perhaps with time they are improving, and I assume that they will continue to improve."

Hard to say, mainly cause he's dead.
This news just in! Bobby Fischer is STILL dead!
(My thanks to Saturday Night Live for allowing me to rip off this joke.)

Bobby with Susan (petting a cat) during the same trip:
Bobby and Susan played a few games together (of Fischerrandom) but those games are not publicly available...
Ok s Bobby Fischer is MAYBE the best chess player ever, and would off course beat any women becase men are better.
Anyway, here's the question: If he played against a woman and he won (off course), and then she asked for a rematch, should he give it?
To me it's a bit pointless that he does since he won already. But then again, it is bad taste in general to refuse a rematch. I dunno.