In the geocentric system, the Moon is still, in a sense, a satalite of the Earth, as is the sun, the planets and the stars as they all orbit the Earth. Aristotle is often credited for the geocetric model, but there are writings that preceed him that describe it. The idea that the Moon orbits the Earth may be even older than that, but I couldn't find anything on that.
I think one of the interesting things the ancient Greeks came up with, was they may have been the first society for whom the Moon was no longer a god. Their gods were human like, not the Sun, the Moon, Nature, etc. The Sun and Moon became objects in the sky instead of part of their mythology. But I digress.
The first written and worked out calculation that the Moon was smaller than the Earth and orbited the Earth as a "moon", and that the Earth was smaller than the Sun and orbited the Sun came from Aristarchus of Samos, who came not long after Aristotle in the 3rd century BC.
If Aristarchus had been a few years earlier, perhaps his analysis would have swayed Aristotle, but that's just speculation on my part. Aristotle's ideas came first and he was well respected and well read.
Ptolomy of Alexandria who'd writings follow about 3 centuries later, knew of both Aristotle's and Aristarchus' models, but he found Aristarchus' model troubling because he thought that the Earth's rotation speed would create high speed winds not seen on Earth, so he leaned towards Aristotle's model and, in addition to that, using epicycles, the Ptolemaic model was highly predictable. It explained planetary motion and people like that.
Predictions, a model that fits what we see as well as the respect for Aristotle's teachings and the Church saying "this is so" may have all played a role in the 15 centuries that the Ptolemaic model endured. We shouldn't judge that as stupid, because there was a logic to it and a lack of evidence to say otherwise. If a system works and can't be shown to be false, it's likely to stick around.
There may have been the occasional scholar who read Aristarchus during the middle ages and agreed with him, and I've read that Copernicus credits Aristarchus, but that wouldn't be enough to sway the accepted theory. Tycho Brahe, for example, who came after Copernicus but preceded and later coincided with Galileo, went back to a revised geocentric model. See Tychonic system where the Moon, Sun and Stars orbit the Earth.
Galileo's telescope, however, provided proof to the contrary of the Ptolemaic system. He discovered Jupiter's 4 large moons, later named the Galilean moons and he observed the phases of Venus and later, he timed the tides to the Moon's and Sun's orbit. Galileo was the first to provide proof that the Moon was a satellite of Earth and that other planets could have moons.