Interesting idea, but unlikely. More likely that Gautama was a family name, and that after Darius seized power the family left, perhaps going to India. Names tend to passed down in families. I supposed the King Gautama could have decided to become a religious acetic after losing power, but that doesn't match our biographies of Buddha.
I could see a prince, a descendant of the deposed king having the same name, having a different slant on the Indian religions of the time due to his different historical background. So while I doubt that they are the same person, they could be related. Wouldn't be the first time a king who lost power when to a different land - some of the Persia princes went to China after the Muslim conquest.