So Newton's experiment won't help in explaining this
No, it does explain it.
First, I'd like to dispel a confusion regarding dispersion. In both cases (slab/prisms), we have no angular dispersion, but we do have lateral dispersion (image stolen from @Amu's post)
The colors do split up, and if you used a thin slit of light, you would get an outgoing slit with coloring at the edges (you may get a rainbow if it's thin enough). For a thicker "beam", adjacent rays overlap when they split, producing white light. But since the rays are parallel, the amount of dispersion does not change.
In a single prism, on the other hand, the rays don't come out parallel--they have an angle between them. And thus the lateral dispersion increases as you go further from a prism, giving a clear rainbow.
In double prisms, depending on the distances, you may or may not see dispersion on a screen.
Actually, the correct experiment is displayed by this xkcd comic:
Yep, there's an extra lens involved--which makes the situation reversible.
Alright, back to the question.
Imagine taking the double-prism in the diagram above and bringing the two prisms closer to each other. The dispersion in the center "z" will decrease till it becomes zero. And then, it becomes the same as a slab--it's as easy as that :). I think that the diagram you have is slightly wrong, I'm not sure--since it's not directly supporting this. I may have to look for some more diagrams and compare.
Remember that a zero-width film of any material has no effect on a ray of light, provided that there's no total internal reflection going on.