International Hydrographic Organization, or IHO, once passed Japan's application of using 'Sea of Japan' alone, but there are problems to this. First, South Korea was under Japanese occupation during that period. Also, Japan was victorious country of the World War 1 then, which means its influence was huge. No other country could have opposed to Japan's application, and this is unjust.
Furthermore, international rules suggest that names of seas shared by two or more countries should be determined through consultation between the countries involved. Since 'East Sea' represents not only South Korea bur also the whole continent of Asia, it is right to notate both 'East Sea' and 'Sea of Japan' to describe the sea.