No, the US government does not own the international court, because in actuality, the US does not have a great interest in it. The US government does not host the international court, nor is most law based on what the international court does. Rather, Europe sets the standard for the international court, and is much more influential.
This question may have had a different answer fifty years ago after the end of the second World War. Currently, with the advancement of military technologies in first world nations, any of a number of countries could effectively devastate the United States in warfare. This decreased the United States ability to own the international court. If that is not enough, the United States debt crisis and China's purchasing of that debt has all but effectively eliminated what once could be argued as the United State's dominance of foreign affairs. With the United States no longer the financial juggernaut or the ability of other countries to effectively strike the U.S., it has lost most of it's power in the international court creating a more equal court.