Just prior to Sammy's use of this phrase, he has watched his boss, Lengel, confront Queenie and her friends about their failure to adhere to the dress code of the store. As Sammy rings up the girls' purchase, he grows increasingly resentful that his boss has insinuated that the girls are not "decent" enough to shop at the A&P.
Deciding to stand up for Queenie and her friends, Sammy announces that he's quitting, hoping that the girls will hear him and find his actions "heroic." Unfortunately for him, they keep walking, leaving him to explain himself to Lengel.
Lengel asks Sammy to repeat what he just said, and Sammy sticks by his decision, asserting that he is quitting. He adds that his boss didn't need to embarrass the girls, and Lengel replies that "it was they who were embarrassing us."
In response, Sammy offers an old saying of his grandmother's: "Fiddle-de-doo."
This is a sign of exasperation that basically means "big deal." It is meant demonstrate that Lengel is being old-fashioned and ridiculous for asking the girls to put on more clothing to simply purchase one jar of herring snacks. It is a form of insult in this context as Sammy separates himself from the societal constructs that have shamed the girls. Of course, Sammy didn't feel at all embarrassed by the girls' scant clothing, and this particular phrase captures his belief that Lengel is uptight and narrow-minded in his views.