Whether a story with zombies is Science Fiction, Fantasy, or Horror, depends much more on what the story is about than how the zombies were created, or the setting for that matter.
For instance, I'd argue that generally a Post Apocalyptic story will always be Science Fiction. But if the main purpose of the zombies in such a setting is to create a scenario where a group of characters have to fight for survival against all odds and are picked off one by one, then you would be into the Survival Horror genre. Survival Horror is mostly a game genre, the Resident Evil games started as Survival Horror (, although they've moved into Action), the first movie is also clearly Survival Horror. (The rest of the movies can be debated much more.)
Generally though, there's a lot of blending over of genres. I've seen Post Apocalyptic Horror been used as a genre. And if you move to movies, Alien is clearly Survival Horror when it comes to story, but since it is has a very clear future setting most people think of it as Science Fiction. -Aliens however is much more clearly a Science Fiction movie, but it still retains much of the Survival Horror from the first movie.
The debate over SFF (, which I use instead of Speculative Fiction since some people still take offense to that term because of its origin,) subgenres is endless. The three main genres blend a lot. What main genre a subgenre belongs to can be the subject of debate, and what subgenres applies to a story is almost always up for debate. Taking Steampunk, I'd argue that its main subgenre is Alternate History, it can't work without it. Alternate History is a Science Fiction subgenre, although you'll find people arguing against that because they say History isn't a science. But since many Steampunk stories contain some sort of supernatural element, many people classify it as Fantasy.
In short, SFF fans have lots of opinions, and like to debate them.
But really, none of what I wrote above really matters. In the last few years I've seen the term Zombie Fiction being used. So if you have a story with zombies, just write it, and let the agent/publisher worry about what genre to put it in when they accept it. -Which is something of a general advice. Genres, especially subgenres, are more marketing categories than anything else.