DirectX is an API. so is OpenGL. ATI make graphics chips, and obviously - they have to work closely with Microsoft/the OpenGL ARB to make sure new features/extensions on their chipsets are supported in the APIs. However, this has nothing to do with Wii, really.
Nintendo have their own SDK, which from what many developers have said - is more similar to OpenGL (the Wii SDK is meant to be similar to the Cube one, and that was more similar to the OpenGL API than DX). As long as it tells the graphics chip what to do, it is of little consequence what the API is.
Nick and DCK are closest to the mark. :P