(So ya, this got a bit out of hand in terms of length. Please excuse the overly indulgent discussion of Jean Buridan in the middle.)

On the contrary, ancient and medieval authors certainly had a notion of gravity which was integrally related to their understanding of the earth as spherical, it simply wasn't a Newtonian understanding of gravity. Unlike the post Newton understanding of gravity as a force independent of the falling body, ancient and medieval authors conceived of gravity as a product of the weight of a falling (or rising) object (hence we get the term from the latin gravitas, meaning weight). The notion was that all things in the universe had a proper place which they sought to reach. Now, since earth is the heaviest element, it naturally tries to amass itself at the bottom, ie. centre, of the universe in a uniform manner. Whereas, on the contrary, fire, being lighter than air, always tries rise above the air. This is why, if we accidentally dislocate an object from its natural position, it will be drawn to its natural position. Hence, things composed of mostly earth and water tend towards the centre of the earth whereas things made mostly of air and fire tend away from the centre of the earth.

The problem of centres is an important point for Ancient commentators. Indeed, when Plato introduces this topic in his Timaeus, this is how he contextualises it:

The nature of the light and the heavy will be best understood when examined in connexion with our notions of above and below; for it is quite a mistake to suppose that the universe is parted into two regions, separate from and opposite to each other, the one a lower to which all things tend which have any bulk, and an upper to which things only ascend against their will. For as the universe is in the form of a sphere, all the extremities, being equidistant from the centre, are equally extremities, and the centre, which is equidistant from them, is equally to be regarded as the opposite of them all.

And he goes on to explain that the fundamental feature of weight is its tendency towards like:

that the tendency of each towards its kindred element makes the body which is moved heavy, and the place towards which the motion tends below, but things which have an opposite tendency we call by an opposite name. (Timaeus, 31 (62c, 63e))

One of the key ancient texts on this sort of thing is Aristotle's On the Heavens, in which issues of weight and relative position are a key concern. In particular in 2.14, he uses this idea of gravity to explain both the sphericality and immobility of the earth. He argues that, if all things have a natural movement, and, under pain of incoherence, can't have two opposite natural movements, it follows that the earth must be immobile, since the earth is simply the accumulation of all the mass in the universe which tends towards the centre, it would require a greater force than that totality of mass to move it, which is absurd:

For a single thing has a single movement, and a simple thing a simple: contrary movements cannot belong to the same thing, and movement away from the centre is the contrary of movement to it. If then no portion of earth can move away from the centre, obviously still less can the earth as a whole so move. For it is the nature of the whole to move to the point to which the part naturally moves. Since, then, it would require a force greater than itself to move it, it must needs stay at the centre.

This also proves that the earth must be spherical since the sphere is the only shape in which the extremities are all equidistant to the centre. Likewise, were the earth unequally distributed, it would then shift so that its centre of gravity matched the centre of the universe:

The earth, it might be argued, is at the centre and spherical in shape: if, then, a weight many times that of the earth were added to one hemisphere, the centre of the earth and of the whole will no longer be coincident. So that either the earth will not stay still at the centre, or if it does, it will be at rest without having its centre at the place to which it is still its nature to move. Such is the difficulty. A short consideration will give us an easy answer, if we first give precision to our postulate that any body endowed with weight, of whatever size, moves towards the centre. Clearly it will not stop when its edge touches the centre. The greater quantity must prevail until the body's centre occupies the centre. For that is the goal of its impulse. Now it makes no difference whether we apply this to a clod or common fragment of earth or to the earth as a whole. The fact indicated does not depend upon degrees of size but applies universally to everything that has the centripetal impulse. Therefore earth in motion, whether in a mass or in fragments, necessarily continues to move until it occupies the centre equally every way, the less being forced to equalize itself by the greater owing to the forward drive of the impulse. (Aristotle, De Caelo, 2.14)

This is also, incidentally, why Aristotle felt the need to

But this idea of a centre to the universe is at least a bit mysterious, and it is central to classical criticism of the sphericality of the earth. Thus Lucretius' argument against the spherical earth centres on the ridiculousness of this idea of there being a cosmic centre:

And in these problems, shrink, my Memmius, far / From yielding faith to that notorious talk: / That all things inward to the centre press; / And thus the nature of the world stands firm / With never blows from outward, nor can be / Nowhere disparted- since all height and depth / Have always inward to the centre pressed / (If thou art ready to believe that aught / Itself can rest upon itself ); or that / The ponderous bodies which be under earth / Do all press upwards and do come to rest / Upon the earth, in some way upside down, / Like to those images of things we see / At present through the waters. They contend, / With like procedure, that all breathing things / Head downward roam about, and yet cannot / Tumble from earth to realms of sky below, / No more than these our bodies wing away / Spontaneously to vaults of sky above; / That, when those creatures look upon the sun, / We view the constellations of the night; / And that with us the seasons of the sky / They thus alternately divide, and thus / Do pass the night coequal to our days, / But a vain error has given these dreams to fools, / Which they've embraced with reasoning perverse / For centre none can be where world is still / Boundless, nor yet, if now a centre were, / Could aught take there a fixed position more / Than for some other cause 'tmight be dislodged. / For all of room and space we call the void / Must both through centre and non-centre yield / Alike to weights where'er their motions tend. / Nor is there any place, where, when they've come, / Bodies can be at standstill in the void, / Deprived of force of weight; nor yet may void / Furnish support to any,- nay, it must, / True to its bent of nature, still give way. / Thus in such manner not at all can things / Be held in union, as if overcome / By craving for a centre. (De rerum natura, 1.1052-82)

But the Lucretian notion did not survive antiquity. Rather, both the sphericality of the earth and the notion of natural movement towards proper place were adopted into the middle ages more or less universally. But this is only really the beginning of the story. In particular, Aristotle's discussion of the matter received no end of discussion. Just to finish up, I'll point to a few points of discussion that came up in the late medieval universities.

I'll deal first with two issues discussed by one of the most prominent late medieval arts masters, John Buridan (~1300-58).

First, on the movement of the earth. Buridan approaches this problem through the question of whether the earth is actually the centre of the universe. As part of his discussion he nicely recapitulates his understanding of the Aristotelean mechanics of the problem: (nb. when these they say 'world' these authors normally mean what we would call the the universe.)

For we suppose that the place [designated] absolutely as "upward", insofar as one looks at this lower world, is the concave [surface] of the orb of the moon. This is so because something absolutely light, ie. fire, is moved towards it. For since fire appears to ascend in the air, it follows that fire naturally seeks a place above the air, and this place above the air is at the concave [surface] of the orb of the moon; because no other element appears to be so swiftly moved upwards as fire. Now the place downward ought to be the maximum distance from the place upward, since they are contrary places. Now that which is the maximum distance from the heaven is the middle of the universe. Therefore the middle of the universe is absolutely downward. But that which is absolutely heavy – and earth is of this sort – ought to be situated absolutely downward. Therefore, the earth naturally ought to be in the middle of the universe or be the middle of the universe. (Grant, 502)

[Cont. in reply]

When do you think gravity was discovered?

For example in about 500 BCE Empedocles formalised the Greek Elemental theory supposing that all matter comprised 4 elements - Earth, Air, Fire, and Water.

Around 350 BCE Aristotle formalised the concept of the Elements having a natural order or place. Things fall because they have Earth as a constituent, things rise when they have Air as a constituent because, Aristotle supposed, the Earth belongs below and the Air above and wrote of it in On The Heavens

Various Ancient Greeks considered our planet to be a ball (in the 7th-4th centuries BCE various theories were espoused). Thus they had both a concept of gravity - substances containing more Earth move more readily to the natural place where Earth lies (not that far from Newtonian gravity) - and of the shape of the planet.

Thus in "Greece" the theory to support observations of a spherical planet Earth were largely coterminous with a gravitational theory, or at least were formalised in and around the lifetime of Aristotle.

I've not studied other philosophies so can't tell you what the Aztecs, Sumerians, Australians, or whoever, thought about it.

See eg "The History and Practice of Ancient Astronomy", James Evans.