You'll probably find this thread from a couple of months back of interest, where I and /u/TheTruthKeeper talk about what people thought about gravity in connection with round-earthism in antiquity.

The only thing I might want to wind back from what I wrote there is that using the word 'gravity' may have been misleading. Aristotle and others of his time thought in terms of universal centripetal and centrifugal motion; they didn't think in terms of F = Gm1m2/r^(2). I'm sure some modern viewpoints would have it that only the latter is 'gravity' per se.

As for the question in your title, we don't know how people reacted, because we don't have testimony about the discovery. We have absolutely no indication of any evidence-based debate about it, so whatever the exact reasoning used, it must have been very persuasive to overturn Anaximander's cosmology. To summarise the other thread, I suggest four key stages in the thought of 6th-4th century BCE Greek observers:

1. Anaximander, early 500s: flat earth, but suspended in space (maybe by air pressure underneath).
2. Archelaos, mid-to-late 400s BCE: still flat earth, still suspended in space, but also with notions of universal centripetal and centrifugal notion creeping in.
3. Some unknown astronomer, ca 400 BCE: spherical earth; primarily based on astronomical observations, but we don't know the specifics.
4. Aristotle, late 300s BCE: spherical earth; heavy things like earth and water move centripetally towards the centre of the cosmos, light things like air and fire move outwards; the apparent circular motion of the stars (fixed stars, planets, sun, and moon) is a non-trivial problem.

As I said above, we have no indications of anyone trying to contest the spherical earth model based on empirical evidence (there are some contrary doctrines from ancient Epicureans and Christians, but not consistently, and not based on anything empirical). It's a bit unsatisfying to say that they didn't have much of a reaction, but that's what seems to be the case.