I can be pretty sure He didn't. Even if He had a hand in it originally, the story was handed around orally for centuries (try playing telephone sometime) before being initially written down. After that, it was copied and copied again and again and again, and during this copying process, done by men, errors would crop up -- mispellings, dropped words, word changes -- not to mention the changes made by scribes to clarify a certain point, or to emphasize a particular doctrinal point.
If there can be 30,000 errors in only 100 versions of the NT -- and that is less than 2,000 years old, how many more errors could crop up in a story that not only was written down hundreds of years previously, but had been kicking around by word of mouth (with all the errors that could crop up there) for centuries prior to that.
So yes, I can say God did not write Genesis, or at least the version we have today.