Gravity naturally tries to pull loose matter into a sphere — toward a central point. In contrast, a spinning motion tries to fling loose matter away from the center. The two forces act against each other, and in different situations result in different formations.
Most planets (including the earth) are actually not perfect spheres: Their rotation causes them to bulge out at the equator. The spin is “trying” to force the earth into a disk, but is not strong enough to overcome gravity. In a galaxy, stars are very far apart from the each other, so their gravity is not strong enough force the entire galaxy into a sphere, so the spin is able to create a disk. Near the center of the Milky Way though, there is a bulge that is closer to a sphere, where the stars are more densely packed.
Also I’m not 100% sure about the details here, but I believe Dark Matter is supposed to be responsible for the shape of our galaxy as well. My understanding is that there is a large source of gravity in the more “empty” region surrounding the Milky Way, which pulls stars outward, preventing a total collapse of the Milky Way into a sphere.
At the start, everything was a sphere of dust with a random density. The clouds of dust started to pull together because of gravity. Now, it’s not just that individual particles of dust close together pulled and stuck together, but all of the dust everywhere. These means that the localized dust clouds formed into planets and stars, the gravity of the other planets and stars pulled on them as well. This pulls things together into a disk.
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