Could ancient astronomers have proven heliocentrism and, if so, how could they have done so?
You can't ever "prove" heliocentrism! (We can't even today. Thinking that science "proves" things is not at all helpful in doing science.) What we can do is show that a heliocentric model of the Solar System is simpler and explains more observations with fewer assumptions. (E.g., a heliocentric model is compatible with Newtonian mechanics and a world where earthly mechanics and heavenly motions are explained by one theory is very persuasive compared with older models of planetary motion. But it is not proof.)
The ancient Greeks and the medievals ran into the same basic problem: They were convinced that motions in the sky must be circular and, in the absence of Newton's work, circular motions around the Sun were no more elegant and no more explanatory than circular motions around the Earth. (One of the main reasons Copernicus' theory was not quickly accepted is that it used circular motion and thus made seriously wrong predictions of the motions of the planets, especially Mars. By around 1600 we had excellent naked eye measurements of planetary motion, probably far better than what the Greeks had. It was Kepler's elliptical orbits which made heliocentrism convincing.)
Remember also that measuring absolute distances to celestial bodies was very difficult without telescopes, so the Sun being enormously more massive than the Earth was not obvious, making its centrality also less obvious.
It's also worth remembering that the focus of astronomy up until the early 1600s was on understanding and predicting the motions of the planets, and not understanding them as physical bodies. (Remember that even in the 1800s some reputable scientists doubted that we would ever be able to understand the true nature of the planets and stars -- then spectroscopy was developed.)
On the other hand, the Greeks knew about ellipses, so a clever Greek mathematician could have proposed elliptical orbits (for Mars and the Moon, at least) and come up with a theory which predicted their motions very accurately. But until you know the relative masses of the Sun and Earth or start to have an inkling of Newtonian dynamics and gravitation, it's really not terribly important which body is at rest.
Bottom line: There was just too much science yet to be developed for the Alexandrian Greeks to find a heliocentric Solar System to be compelling. (But even then, never proven.)