If it were true that the solar system is an atom in “very massive material”, then this “matter”, would need to contain “atoms” that are quite different from ours.
First, a lot of high school textbooks depict atoms as mini solar systems, with the nucleus at its centre and electrons orbiting on a plane similar to planets around Sun. However, this is a very simplified and even wrong picture.
Anupum is correct in stating that actual atoms contain electrons in a 3 dimensional space. This is not just an exisiting on the plane. However, the 3D model he presented is not the right one. Real atoms have electrons that orbit around the nucleus, not like particles moving in random directions. This is the Bohr classic model of an Atom, and it has a serious flaw.
The fact that electrons travel in circles around the nucleus and are charged (negatively) and accelerating (centripetally), would mean that they radiate energy in form of electromagnetic waves. According to the laws of electromagnetism, accelerating charges must emit EM radiation. The electrons lose energy through radiation and slow down. They fall closer to the nucleus. This is a problem because eventually, the electron will spiral into the nucleus and collide. This would destroy the atom. If atoms were really like solar systems, then we wouldn’t be here at all. (The reason actual solar systems don’t have this problem is that planets are made of gazillions upon gazillions of atoms. The charges cancel each other, making them electrically neutral, and they won’t emit EM due to centripetal acceleration).
What does an actual atom look? To answer this question, we need to look into quantum mechanics. Quantum mechanics holds that particles can have wave-like properties. Our electrons can behave as waves. Instead of moving around the nucleus in little balls, electrons can behave like stationary waves. (If you don’t understand what a stationarywave is, try stretching a slinkyspring out and shaking one side, while keeping the other still. The wave movement that you’re creating is a stationarywave. Imagine the electrons curled up into a loop. An electron is like a cloud. It exists all around, but only in the area of its loop. These stationary waves, also known as “atomic orbitals”, are actually 3D. They have many interesting shapes, such as spheres or dumbbells. I recommend that you search google images for them, they can be difficult to describe in words. As you can see, the wave image of an atom is quite different to a solar system.
I hope this answers your question.
Yours is a fascinating philosophical idea, though. Although we can say that the solar system does not look like our atoms, and future advances in astronomical observation may even reveal whether these structures are greater than our universe’s solar systems, galaxies, or super galactic clusters, we still need to know how we can prove that our universe is not a part in some higher degree of order where we behave like atoms in very massive matter. This would be beyond the scope of science. I think we will have to leave this one to the philosophers.