adding to this: There's no airside pet relief areas at ICN. You will have to exit the terminal for a proper walk (there's trees and grassy areas in the parking lot right across from the arrivals level).
If you aren't originating in Korea, you'll need proper paperwork for him to enter the country. The requirements are pretty much standard:
http://www.airport.kr/iiacms/pageWor...de=C1202010700