Definitely no godfather, but a very important figure and a transitional one. Aside from his movie career, as a martial artist and philosopher he did do a lot to transcend the barriers between the martial arts.
But I think that for the roots of MMA you have to look back to the days of real professional wrestling (e.g. non-staged) in the 19th and early 20th centuries. Modern MMA basically comes from Brazil, but there was a vibrant fight scene in Brazil going way back. It was in this context that a disciple of Jigoro Kano (founder of judo) traveled to Brazil and trained the Gracie family, for instance (who from thereafter continued to cultivate their own style). The Gracies are actually responsible (at least in the immediate sense) for the advent of MMA in the US in the modern era. There were other things, like the Gene LeBell-Milo Savage match, and even the Antonio Inoki-Muhammad Ali match. But still, with the founding of the UFC the Gracies basically transplanted vale tudo from Brazil to the US. It additionally grew up in Japan, originally as a sort of appendage of pro wrestling (hard style).
But Bruce Lee is an important figure as well. Not just for popularizing martial arts, but also for his approach to them (i.e. his philosophy of simply using what works regardless of school affiliations, which is reflected in his own martial arts philosophy of jeet kun do).