The only case you can have dual citizenship is if you acquired them by birth. So it doesn’t apply to you.
You have two choices.
1) Do military service. They will arrange time off so that you can travel to the US and keep your green card valid. Unless you really need the US citizenship, you can live as a green card holder forever. You will still be Korean citizen and can go back any time.
2) Acquire US citizenship. Live as non-Korean.
If I were you, I would pick 1).