No, the Intelligent Design Theory does not prove the existence of a God as Intelligent Design is simply an off-shoot of the philosophical idea of Deism, which in turn does not soundly prove the existence of a God. As God cannot be viewed empirically, the existence or lack thereof of God cannot be proven in a scientific sense.
In fact, the world is very poorly designed, as are humans. We all break down, get cancers, suffer from multiple diseases. It's been man's CREATIVITY that has led to medicines and other inventions which have made life bearable and better. The Universe itself is filled with 90% dark matter and is mostly dull and lifeless.