Although Galileo contributed greatly to science, his work also had impact on history. Galileo's discoveries helped to diminish the role of the church as the ultimate authority on the way the world worked, which in turn opened the way for more scientists to counteract the church's (often inaccurate) views about the natural world. In some ways, Galileo's led to the era of history called The Enlightenment, which was a break away from the superstitious and religious beliefs of the past (for example, astrology) in favor of a more scientific-oriented approach (for example, astronomy).
In history classes, teaching about Galileo would put him in the context of the politics of the time, demonstrating how people can be treated for expressing views that diverge from convention. In science classes the focus would be on his discoveries and their meaning and impact on how the universe is viewed in relation to ourselves.
I believe that Galileo should be taught in both history and science classes. Generally, I have found this to be the case, but for schools that don't, they should consider it. Galileo worked in science and became a prominent historical figure due to his problems expressing his findings, which were later validated.
In history class they should focus on Galileo's historic achievements as he played a major role in the scientific revolution and also his battle against the church during the Roman Inquisition. Scientifically there is also a plethora of areas that could be studied about Galileo such as all of his work in astronomy and physics.
No, I do not think that Galileo should be taught in both history and science classes, because his contributions were more to science than to history. There are more notable people in history to discuss. Galileo's contributions to gravity and other fields of science are best saved for science. He is not a notable person without his science.