I am writing a science fiction story (actually a screenplay for a film, but I imagine that for the purposes of this question the same rules apply) and I have been told that it isn't really science fiction. It's a series of vignettes with an overarching plot about the snowball effect, and is designed to not really have any single antagonist; the primary "vessel" for this is social media and its abuse. All of the social media used does currently exist (it's a film for a school project, so I get fair use). Does this... count as science fiction? If not, why?

It isn't science fiction, it is just Contemporary fiction, aka Realistic fiction.

A Science Fiction story must rely heavily on some non-existing tech or some reasonably plausible guess at a futuristic development; like being visited by aliens, or discovering them.

In some scenarios (about the future, or space operas, etc) there is a lot of this; in others it can be just one thing; e.g. a time-travel story may have only one piece of non-existing tech (the time machine).

The NEW tech (or discovery) must be central to the story line and what happens, e.g. in Star Wars, FTL space travel, light sabers, sentient robots and sentient aliens are all central to the plot. In Men In Black, only a little new tech is important, but aliens are central to the plot.

It isn't "science fiction" if it is about science but the science employed is not fictional; it is what we already know, you offer no surprises or eye-opening innovations on the tech front, or about the future.

Social media is no longer sci-fi

In years gone past concepts like the internet would be considered sci-fi. Many book exist which use digital communication on a global scale as part of a sci-fi setting. Google defines science fiction as:

fiction based on imagined future scientific or technological advances and major social or environmental changes, frequently portraying space or time travel and life on other planets.

20, 10 or, to some degree, even 5 years ago it would have been difficult to imagine a single network or service having as much control over our lives as facebook does today. A setting that used such a network would have been considered sci-fi or at least dystopian as little as 5 years ago, but wouldn't be today.

For example consider the novels Feed (2002) and The Circle (2013) both books deal with an overarching network connection with control over our daily life, set in the not too distant future. Feed is a distinctly sci-fi novel, written when the internet was still relatively new and the idea of a network like that is a major technological advance and social change. The Circle is slightly less sci-fi and more dystopian or alternate reality as the technology and social structure are less distant from reality at the time of writing.

Move forward another 6 years to today. The concept of a social network is not longer futuristic and required no technological advances or social change. They already exist and writing on them would be more closely considered social commentary than alternate reality.

In conclusion; it would have been sci-fi if you wrote it 10 years ago. Today it is just fiction.