As a somewhat average person who doesn't frequently get into deep discussions of racism, I'm one of those who would instinctively define racism as bigotry due to race. Institutional racism is a different thing. All the -isms: (sexism, ageism, anti-semitism, etc) bigotry due to -whatever-. It's putting "institutional" before the word that makes it about something more than one individual bigot's words, beliefs, or actions. At least, that's how I'd define it.
OP - no, I wouldn't have done anything. I kind of feel like bigoted people are entitled to their bigoted beliefs, it's when they act on those beliefs in a way that causes harm to others that it becomes a problem. The woman's comments weren't directed at you (she may not have even known you, a white person, was within earshot - I have no idea).
I *guess* there's the argument that expressing bigoted beliefs harms people, especially impressionable children, and may pass those attitudes on, but... eh... We do have freedom of speech and she's entitled to voice her belief that she hates white people if she wants to. It may, arguably should, result in people shunning her as a result, because being bigoted is a bad thing, but it is her right to say that's how she feels. We could get into a debate about times when the "freedom of speech" argument is being taken too far - for example, a black boss saying to their white employee, "I hate white people." There's more at play there than a random person expressing their beliefs. Whereas a random dude walking down the street commenting to the guy walking along with him, "I hate white people," well, that's really nobody else's business. Fine, you hate white people, I'm going to make a concerted effort to never engage with or befriend you, Random Dude, because I don't like bigoted people.
Talking during a movie, on the other hand, that's just generally not allowed. Every theater I've been in tells you to not talk during the movie. But regardless of what the comments were, I still personally wouldn't say anything. You usually have to be so disruptive that the majority of people around you are having a problem with it before it seems escalation to theater staff is appropriate.