As I see it, there are two type of anger range. 1, Tampo - a mild disagreement where you cannot agree and your wife refuses to understand your point of view, but resolved with a bit of loving after a day or two. 2, Sumpong - a total refusal to see your point of view and complete refusal to see any logic or resolution to the problem.
I think you are at number 2.

Of course we only know a brief bit of the argument and it must be hard to have this in front of a young child, this must be avoided at all costs in my opinion.

It's difficult to know what to suggest, but try and make her see some sort of reason, some sort of logic and find a resolution that you can both agree on. If you have to say sorry, then do it. The main point is that your child is not the loser and she still sees that you are strong and love each other. We all go through tough times and the cultural differences still remain no matter how many years you have been together. Try and explain how it is affecting you and the child and find a solution.