I really didn't want to read the whole topic before responding, so I'm problably going to say something stupid..
But.. Greek fire was invented in ancient Greek times, right? Wasn't that around 3000 years before the Islam even existed? So, why would anyone go back to 700 AD to invent Greek fire while it was already invented?
But, eh. Time traveling doesn't seem that impossible. To get back on topic

EDIT: What the hell, I thought it was invented 3000-4000 years ago. Well, my history lessons slacked, appearently it was used around 500 AD for the first time. Says wikipedia. So the islam still didn't exist.
Just so we're clear on our chronologies, even if the ancient greeks invented Greek Fire, it still wouldn't predate Islam by 3K years. Even Bronze Age Mycenaean Greece didn't come along until ~1500BC, putting it maybe 2K years before Islam, and if by Ancient Greece you mean Classical (and you probably do) we're talking more like less than a thousand years. 3 thousand years prior to Islam would put us in the very dawn of Aegean Civilization.