Not really. It was probably similar to naphtha or naphtha was a component. Interesting topic, I wouldn't mind knowing the true answer.
It's not a secret. I have a book of formulas from the 30's which gives the recipe. Perhaps the actual recipe that the Greeks used is a mystery, but the same effects can be replicated with modern chemicals.
Wait, I have it. The Colonel's secret recipe. There, will that do? He's probably just guessing though, obviously.
"Greek Fire is made as follows: take sulfur, tartar, sarcocolla, pitch, melted saltpeter, petroleum oil, and oil of gum, boil all these together, impregnate tow [i.e., the coarse or broken part of flax orhemp, prepared for spinning] with the mixture, and the material is ready to be set on fire. This fire cannot be extinguished by urine, or by vinegar, or by sand Anyone wanna explain to me why they tried urine and vinegar before they tried WATER!
They tried water first but it made the fire spread even more. Actually, what I found in -at least two- greek sites was that vinegar and urine did extingush the fire. I also read that the exact nature of greek fire is not known even today only some substances mentioned by the other members. That is because the officers (in Byzantium) responsible for the production would talk about it to the emperor only and no one else. In Greek it is called liquid fire -igron pyr. The English called it Greek fire when Richard I found the secret formula in about 1194.
Part of the legend one learns at Johns Hopkins concerns one of JHU's great early physicists, R W Wood. It is said that occasionally after rains, he would go into the slums of Baltimore with some small chunks of Sodium (under oil, of course) and toss them into puddles. (I do not recall if that alone is enough to cause the water to appear to burn as hydrogen is released or a lit match was tossed in soon afterwards.) Legend has it that he dressed appropriately in black etc.