I think it was not napalm alone... According to my organic chemistry textbooks, the polymeric substance that formed napalm was not invented in Greek times, there should be more logical answer:

My Proposal for Greek Fire:
Greek fire was made from a mixture of naphta(or other aromatic compound), tar(for filling), and nitric acid/HNO3 (made by mixing saturated vinegar with saltpeter(NaNO3)... That mixtures was put in a wax tube-case, and transported as a solid ammunition. However, prior to their usage, they are melted down in situ (in their tube-sprayer), and ignited soon before spraying...