I can talk about the Greek's account since I'm well read in ancient greek philosophy. You likely heard about the four/five elements idea of the world - air, water, fire and earth, plus aether. This, similar or various permutations were common accross the classical world - I'm not much knowledgeful about how the Indian or Chinese versions of these exactly worked, just aware that other civiliziations employed similar theories too (hopefully someone else can shed light on them).

So as far as ancient Greeks go, we know that the idea that certain elements make up the material world goes as far back the pre-Socratics, starting with Thales of Milethus, regarded as the first Greek philosopher. He personally held water was the primary substance from which everything else derived. Then you got Heraclitus who held fire in high regard, and for Anaximenes it was air. Now I don't want to go in-depth about these guys, though as little and fragmented we have left from them it's quite fascinating how they toyed with ideas, trying to establish sense in the world, as this would go far off-topic. Let's just stick to the part they recognized that nature can be reduced in some ways to essential substances (but I need to stress only some thought so, others had quite different ideas - there was Pythagoras and his followers for whom it was all about numbers, or Parminedes and Zeno whom saw the universe as an unchanging reality).

Where this element thing got it's form that held up in some versions till the late middle ages was Empedocles - he wrote that there are 4 primary rhizomata, roots as he called them, water, air, fire and earth, plus two ordering principles, love and strife. These substances are indestructible and unchangeable, the only thing that varies is their share in composition which produces the seemingly infinite plurality of stuff we see in the world. In an usual talk about Greeks natural philosophy we'd now go into accounts like the atomism of Leucippus and Democritus, but for the purposes of answering your question I don't think its necessary to go that way. Moving on, Plato likely picked up the idea from Empedocles and gave these substances the name stoicheion, elements. At least as far as Aristotle tells us, since no works survived from Plato that tells us about these ideas, so take it with a grain of salt.

So let's move on to the real show, Aristotle. In his work On Generation and Corruption, he lists the four usual element, and assigns each two quality out of four. So earth is dry and cold, fire is hot and dry, and so on. Then in On the Heavens he adds aether, since he perceived the four elements as material ones, too earthly and corruptable for the spheres above us. Our terrestrial sphere is made up of the original four, and the heavens above (crystal spheres as he calls it) are made from unchangeable aether. Since it's unchangeable and the only movement the heavens do is perfectly circular (and so he explained the apparent movement of stars and planets), everything else thats ephemereal is made up of the terrestrial elements (this has some quirky implications, like comets and asteroids, but even the suns heat belonging to "us", not to the sphere of stars). For Aristotle, our lowly elements are not unchangeable, but can transmute to another they share a property with (say hello to alchemy).

You may be wondering this is all nice, but what does this have to do with ancient accounts of gravity? Well Aristotle answers that too: Each elements has a natural place - earth is usually in the bottom, so that must be it's place. Water stands above earth but below air. Air is above both; one can observe how air bubbles rise up from water if trapped. Fire always "wants" go up, leaving even air behind. Aristotle does say this difference in resting places manifest in the weight of elements, earth being heaviest and so on. These elements formed concentric shells (except earth which is a sphere), each lighter than the other below it, until we reach the rarified aether. And when an element is displaced from its place it will seek to go back. Earth will fall down, water will flow down too but stay above earth, air and fire will rise up, and aether is always above us in heavens, moving only in circles. There you have it, a model on gravity and a bunch of other physical phenomena like bouyancy, phase changes, hydrodynamics, celestial mechanics and so, elegantly explaining the workings of the world based on relatively simple principlies and substances. Of course if you start to do detailed poking you will find some irregularities and exceptions, but to be honest greeks didn't concern themselves with these, they hardly carried out experiments or did field tests - to them, a philosopher had no business meddling with nature, material and corruptible it was, only by their intellect they can deduce any truths, least it comes into contact with physical world the better. They recognized they were forced to trust their senses having no better avenue (though for some sceptics this meant one mustn't make any dogmatic conclusions).

This explanation on its own might strike simplicistic ("duh', earth falls down, that's it?"), thats why I went through the short history of Greek theories on nature, so you can see how this all adds up and why in many ways it made great sense of nature and explained much of how the world works and why it persisted for so long.