Well, the thing about atoms is that they arrange themselves in patterns to create those larger building blocks you speak of. Solar systems move much too slowly, so that by the time they would have arranged themselves into anything resembling the patterns exhibited by atoms, the heat death of the universe would have occurred.
The resemblance you see is orbit, but the major issue with uniting the orbit of atoms and the orbit of planets under one theorem is the scale of the forces at work. Gravity is many orders of magnitudes weaker than electromagnetic force holding electrons in place (and it needs to be that much stronger because of how much faster electrons move relative to their size than planets).
But, that assumes that it is the same time horizon as what we experience. I've often thought of the scenario where we are living on something that is like a quark of an atom of a much larger existence. And that maybe we make up a small bit of a dude who is sitting around having similar thoughts. But, this enormous (to us) dude moves on an extremely slower timeline. So slow that they there is no possible way we could communicate, even if either of us realized. And in the same regard, there are whole worlds that exist inside the quarks that make up the atoms that make up us. Maybe it is even a recursive existence that goes on to infinity. Because, why not? Prove me wrong. :-)
Subatomic particles are still constrained by the same speed of light as larger objects. As you scale up the speed by which this recursive universe operates in, this limit becomes more and more significant, and fewer interactions can occur in the relative unit of time.
To put it another way, if this super-universe were to use solar systems as atoms, the speed of light would mean their timescale would be in the billions of our years to their seconds. This is derived from the picosecond delay of forces acting between our atoms and scaling up to the solar system "atoms" that make up our galactic neighborhood (10-100 light years apart). So solar systems couldn't be atoms on this timescale because they would do little but coalesce some of the intergalactic medium and die in seconds.
It's possible, but the theory assumes we're operating within the same physics, just different scales of time and space. Supposing there are other universes with their own laws of physics is rather arbitrary, and you could literally argue anything :)
I would argue a universe as a unit is a terrible candidate for an atom for a super-universe since our physics assumes it is a closed system. It would be neat if we weren't bound by the heat-death of the universe and somehow low entropic states could leak back in. But that is all pure speculation and it cannot be proven or disproven from a scientific point of view.
The solar system and many like it are already part of a galaxy, and multiple galaxies form a galaxy cluster or filament due to gravity. Overall they already formed a neural-network-like pattern.
Great point. We should also add for @Vupperware 's benefit that subatomic orbitals the way they're envisioning them are a lie we tell ourselves because quantum mechanics is too damn weird to think about.
In fact, probably the greatest argument against atoms as smaller scale worlds is the fuckiness (technical term) of quantum mechanics on that scale. "Worlds" existing only as a probabilistic distribution might make existence difficult.
See I'm the opposite, the fuckiness at that scale is my greatest argument for their possible existence, accepting that their existence would be in a manner completely alien and unintelligible to me. There's SO MUCH fuckiness that anything is possible.