In a typical day on the NYC subway I am likely to hear half a dozen languages other than English, some from immigrants and their families, some from tourists.
I know a lot of people want to kick the UN out of NYC, but it's got nothing to do with the languages spoken there.
I find it ironic that countries like Canada, Belgium, Cyprus, etc. embrace their multi-linguistic culture and officially recognize more than one "official" language, but in this country, where we have no "official' language, there are people who get bent out of shape if someone dares to speak a language other than English.
The OP and her family were speaking Arabic among themselves. The other woman was rude to be eavesdropping on their conversation and even more rude to confront them about something that was none of her business.
I do think it's rude to speak a foreign language in a group setting if not all of the members of the group understand that language.
I agree that it behooves an immigrant population to learn the dominant language of our country, but I don't have a problem accommodating those who don't speak English, or don't speak it well.
And I have a real problem with those who complain about Spanish. Why? well, among other reasons, because Puerto Rico, whose people are born American citizens, is bilingual, recognizing both English and Spanish as "official" languages. Imagine that, a territory of the United States is officially bilingual! Friend of mine got a lot of grief when he applied for a passport at a post office in New Jersey when he presented his birth certificate, issued in Puerto Rico, because the document was in Spanish.