Imagine you work in a United States office where some employees are from the USA and others from Korea. What is an acceptable way for an American employee to apologize to a Korean employee when the American believes they have caused the Korean to lose face by criticizing them in front of coworkers in a group meeting?
I'm from Japan whose culture is quite similar to Korea's.
If it was in a US office, I'd honestly recommend treating them just as you would treat an American colleague. It's a nice thought to be culturally sensitive, but most immigrants are perfectly capable of understanding other cultures and adapting after living and working there long enough. If somebody interacts with me in a different way because I'm from Japan, I'll probably be annoyed even if it's well-intentioned.
If the colleague is very new to the US, or are just visiting it might be a different story, but even then you might risk coming across as patronizing. After all, it's not like we east Asians come from a different dimension - most likely immigrants working in a highly-skilled profession will have a decent understanding of the culture they are working in.
One followup question: do I need to make other coworkers aware of the apology? A private conversation does not communicate to the coworkers that were in the original meeting that an apology happened. Is that important for the apology to be appropriate?
Honestly, I wouldn't try. Unless you have a very good understanding of Korean culture, if you do things you normally wouldn't do, chances are you will do it wrong. Better to be sincere in your own way, or perhaps be a bit nicer than normal, but I wouldn't recommend trying to use culture-specific communication tools you aren't familiar with.
I am American but has spent time working in a East Asian country and can understand your view. There is indeed are some differences in how interactions between coworkers work such in your circumstance. Before answering your question, I want to provide some useful background.
Articles on Korean business culture such this one states that communication in Korea is more indirect and anything that may place a coworker in a bad light, especially in public, is usually perceived as bad taste / rude. In Korea, the emphasis on harmony, unity, and collaboration means that one usually tries to speak with one voice and avoid opinions / comments that take a different view. Therefore you colleague may have perceived your public criticism as making him "lose face" or that he had to defend himself, which he may not be expecting or particularly good at, per the cultural environment he is familiar with.
Going forward, if you need to critique or otherwise need to give negative feedback to this colleague, I suggest a private conversation. This will give him an opportunity to correct himself or provide his feedback to you before any public exposure in front of other teams and / or management.