Galileo Galilei did not definitively prove that Earth goes around the Sun, but he provided strong evidence in favor of the heliocentric model, which posits that the Sun is at the center of the solar system, and the planets, including Earth, revolve around it. This model was first proposed by Nicolaus Copernicus in the 16th century. Galileo's observations and conclusions in support of the heliocentric model are as follows:

1. Sunspots: Galileo observed sunspots and concluded that the Sun is not a perfect, immutable sphere, as previously believed. He tracked the movement of sunspots over time, which provided evidence that the Sun rotates on its axis. This observation challenged the Aristotelian view that celestial bodies were perfect and unchanging.
2. Phases of Venus: Using his telescope, Galileo observed that Venus goes through a complete set of phases, just like the Moon. This observation contradicted the Ptolemaic geocentric model, which predicted that Venus would only exhibit crescent phases. The heliocentric model, however, could explain the full set of phases, as it posits that Venus orbits the Sun, and Earth observers see different portions of its illuminated surface depending on the relative positions of Earth, Venus, and the Sun.
3. Moons of Jupiter: In 1610, Galileo discovered four moons orbiting Jupiter, which are now known as the Galilean moons. This was a significant observation because it provided the first evidence of celestial bodies orbiting around an object other than Earth. This discovery weakened the geocentric model and supported the idea that objects could orbit other centers, like the Sun in the heliocentric model.
4. The Milky Way: Galileo observed that the Milky Way, which was thought to be a nebulous cloud, was actually composed of numerous individual stars. This observation suggested that the universe was much larger and more complex than previously believed, further undermining the geocentric model.

While these observations and conclusions did not prove the heliocentric model beyond any doubt, they provided substantial evidence that challenged the geocentric model and supported the idea that Earth orbits the Sun. It was only later, with the works of astronomers like Johannes Kepler and Isaac Newton, that the heliocentric model became more widely accepted and understood.
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