I was once walking to a restaurant with my Korean friend when we came upon a husband and wife arguing unusually loudly. Suddenly we looked over and saw the man start punching his wife (a tiny 20-30 something girl) in the face numerous times. Instinctively, I ran over and shoved the guy off her and held him back. I had to fight the urge to kick his ass so I wouldn't get in trouble and deported. If it weren't for the fear of deportation he would've been knocked out cold.
In the midst of this, I noticed a crowd of Koreans (all ages) standing around and watching...doing absolutely nothing. The woman had blood dripping all over her sweatshirt, which got on me as well, and all these people could do was watch. Even my Korean friend stood (at least in awe; most others watched like it was entertaining). Only after I broke it up and got the guy to calm down did someone come over and call the police.
Oh, and did I mention that this happened right in front of the couples' two little children of about 6-7 years old? Yeah, even then, no one cared. Not one person came over to the woman to help and see if she was okay. The woman simply walked home from there, holding her kids' hands, while bleeding on her shirt and sobbing. It was possibly the saddest thing I've ever witnessed in my life.
I mentioned this to my adult classes the next day and some of them told me exactly what you said: I should've minded my own business because it's "a private matter" or that I can be held accountable if the guy blamed me (god knows how or for what). Others said that they wouldn't do anything because they'd be worried about the inconvenience of having to wait around, talk to police, etc.
The complete lack of concern for anyone who's not friends or family is one of the biggest problems in Korean culture. It's utterly disgusting.