He is your advisor but he doesn't own your life. – Nate Eldredge
Professors/supervisors do not own people like their pet turtle. – Greg
Your adviser is a jerk, regardless of culture. This is 2015, not 1895. – gnometorule
I don't think there is such a thing as "regardless of culture". The world is very diverse and sadly, the English-speaking Internet tends to forget that.
As you may have noticed, the hierarchy is very strong in Korean society. Your professor may not "own" you, but he is both older than you and hierarchically above you. As such, you owe him respect. In theory, if he gives you an order, you have to comply, whether you like it or not. If he asks you to come work on a weekend or late at night, you have to. In practice, rules are more relaxed for foreigners, who are not expected to know all these cultural details, but I suggest you be aware of them to avoid any mistake!
What you really have to remember is that you and your professor are not equal. In all situations, your professor dictates what is right and what is not, what you should do and what you should not.
That being said, there are (mainly) two kind of professors:
Your professor is from an older generation and / or he is a reputed researcher in his field. This kind of individual tends to be very traditional and very finicky on "proper respect". If this is the kind of professor you have, I strongly advise: do not displease him. Just obey. Lie if you have to, but do not go against him in any way, the consequences may get way out of proportion. I am serious!
Your professor is more understanding and used to dealing with foreigners. First, start with an apology and try to explain the situation with him: you didn't know you were not allowed to do what you did, and that you will refrain from doing it without his approval in the future (again, lie if you want to, just try to not offend him). He will probably listen to you (but it's unlikely that he will change his mind anyway) and move on. Perhaps try negotiating a collaboration with him and the other author's supervisor.
So, in short:
How can I mend things with my professor and continue to collaborate with my coauthor?
Make an apology, do (or say you will do) what your professor wants, and continue collaborating discretely. Do not get caught again and do never mention any external research again. If asked about it, feel free to lie if needed. It is OK to say something and do the opposite, the only important thing is to not show disrespect.
This may sound stupid or senseless to foreigners, but this is how things work, whether you like it or not. Hopefully, it will change over time, and in fact, it is already changing, albeit slowly.
Hope that helps.