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I get that Galileo Galilei was a major contributor in proving the helio-centric theory, and same goes to Sir Isaac Newton. The battle for the theory to become law took centuries to prove. People finally agreed that the Earth was not in the center of the Universe. But when?
How long did take for people to finally accept Copernicus’ theory? The time starts on the year Copernicus’ theory was introduced to the world.
Heliocentrism did not start with Copernicus, and Copernicus's theory was not the one accepted, he still had circular orbits and equants. It is Kepler's version that became the consensus after Newton derived elliptic orbits from the inverse square law in Principia (1687). That "proved" it to most scientists. We can count from around 270 BC, when Aristarchus first suggested the idea, or from 1533, when Copernicus reintroduced it in a much more elaborate from, or from 1609 when Kepler replaced circles by ellipses. Wikipedia gives a very detailed account of history that would answer the question for any starting date, and under any interpretation of "prove".
Keep in mind however that under both Newtonian and relativistic physics the question of what revolves around what is subtle. A reference frame can be attached to both the Sun and the Earth, and either one will then revolve around the other in it. We need a way to select a preferred frame, for example according to the simplest description of dynamics. For the two of them that would be the frame attached to the center of their masses (which is inside the Sun because Earth is so much lighter, but not at its center), and they both revolve around it.
The biggest factor in the general acceptance of the helio-centric model was probably Kepler's Rudolphine Tables, published in 1627. These used Kepler's heliocentric system with elliptical orbits* and were a substantial improvement on tables found using the Ptolemaic or Tychonic systems. Their practical use in predicting planetary positions for astronomers, astrologers, doctors, etc. in the following decades was probably the biggest factor in swaying Europe's educated population in favour of heliocentricism.
*(one of the reasons Copernicus's system wasn't accepted early on was that tables created using his circular orbits weren't any more accurate then those found with the Ptolemaic system).
Depends on when people started thinking about it the issue. Modern humans evolved some 50,000-100,000 years ago, so somebody probably started mulling the issue over at that point.
A better way to phrase the question would be, when did humans acquire the observational tools and the math to determine that despite appearances, the earth orbited the sun and not the other way around.
Because the issue got caught up in the propaganda wars of the Reformation, it's easy to think that the movement from geocentrism to heliocentrism occurred because of some shift in philosophy or theology. It didn't. It happened when technology provided better observations. From the invention of the telescope, the matter was settled in less than 50 years. Pretty quick for the times and the ongoing conflicts.
