You could solve the problem by calling zombie movies horror flicks.
If zombie stories have to be either SF or fantasy, you're right that the "science" isn't really there, but the "magic" or "otherworldliness" or whatever it is that distinguishes fantasy isn't present either (except maybe by default if you want to argue that zombies could never have a scientific explanation and therefore have to be some kind of magic).
Other difficulties: 1) the setting is realistic, not fantastic, 2) post-apocalyptic stories have come to be accepted as science fiction (indeed, almost to dominate popular science fiction) and 3) there are a lot of quasi-zombie SF stories -- it's the "rage virus" or something like that that lets you tell something a lot like a zombie story and still call it science-based science-fiction.
Depends. ( Yeah I know the scifi/fantasy/sword and sorcery trichotomy )
Will smith in I am Legend (and omega man and all the prequel/sequel/reboots) pretended to be scifi.....like the virus that killed most of humanity, that must be cured (and made the apes smart in the latest planet of the apes reboot).
The best virus scifi movie is still the original Andromeda Strain. No zombies though.