The Sweden Solar System, at a scale of 1:20 million, is the world’s largest scale model of the Solar System, with the Avicii Arena representing the Sun. The Arena was the largest spherical building in the world… until the completion of Sphere in Las Vegas.
It is clear what America has to do.
Sphere is 516 feet in diameter, implying an overall scale of 1:8.855 million, making the model over twice the size of that of Sweden. Even at this scale, most of the planets, dwarf planets and moons are surprisingly small. Earth is the largest non-gas-giant in the Solar System and would be represented by a sphere less than five feet across, about 10 ½ miles from the Sphere; there’s a convenient casino about the right distance and not far from the road towards LA. The Moon would then be a 15 inch sphere 142 feet from the Earth.
The four gas giants present a more serious challenge; even the smallest, Neptune, must be 18 feet across. While it’s preferable that all bodies be represented with spheres or hemispheres, in these four cases we might initially follow the example that the Sweden Solar System sets with Jupiter and Saturn, and use a circular representation at least initially. For example we could represent Jupiter as a painted circle, 53 feet in diameter, at the Cima Road truck stop. I’ve put the locations on a road trip from Vegas to SF via LA, since that should maximize visitability and also is most convenient for me. Note that the distances increase rapidly; while Mars can be placed near a race track just outside Vegas, Jupiter is already out of Nevada and into California. Uranus can be placed at a botanic garden in Greater Los Angeles, Neptune at a hotel in SLO, Pluto in the Exploratorium in SF, and Eris in Portland Oregon.
One convenient thing about this scale is that even relatively small astronomical bodies will be visible to the naked eye: for example, Mars’s moon Deimos will be a sphere slightly under 1/16 inch in diameter, around 9 feet from the 30 inch sphere representing Mars itself.
This spreadsheet details the diameter and distance each model needs to be to fulfil our chosen scale, and in most cases suggests a suitable location at roughly the right distance from the Sphere.
The relative distances in the solar system, and the vast space between its bodies, can be counterintuitive. Now that America has claimed the title of the world’s largest spherical building, a grand project to make this the center of the world’s largest scale model of the solar system is the perfect opportunity to capture the vastness of space.
Followup: What about Alpha Centauri? Photo credit: Sin City Las Vegas
Minds aren't magic 
Deimos isn’t really a sphere, but at 1/16″ it won’t be easy to see its true shape.
We can 3D print something for anyone who wants to look really closely?
I like the idea of having smaller bodies installed inside fixed microscopes, perhaps with some familiar 1:1 scale objects alongside, like a coin/grain of salt/human hair