One thing I wondered about is whether this idea is plausible at all. I think it's pretty clearly not for reasons I'll go into below, but the initial question is can you make something strong enough to do what you want to do ignoring practical considerations?
In theory
So, first of all let's consider a simplified thing: two equal masses connected by some kind of light cable being spun, and at some point you'll let go of one of the masses (and deal with the other one, and the cable, somehow...) The question is whether you can make the cable strong enough.
Let the masses be $m$, the cable have length $2r$, and the angular velocity of the thing be $\omega$. The masses are moving with speed $v = r\omega$, and the centripetal acceleration is $a = r\omega^2$. So the tension in the cable is
$$T = m r \omega^2 = \frac{mv^2}{r}$$
Let the tensile strength of the cable be $u$, then the strength of the cable $\pi u d^2/4$ where $d$ is the diameter of the cable.
So we can rearrange this to get $d$, which is the interesting thing: we need $d$ to be really small otherwise our approximation goes horribly wrong as the cable is not light and you have to do harder sums.
So in the light-cable approximation then you get:
$$d \ge 2v\sqrt{\frac{m}{\pi r u}}$$
(I have convinced myself that this is OK dimensionally, anyway).
So, let's assume you want to give something escape velocity, and you're going to use carbon nanotubes to make the cable. Let's assume:
- $m = 1\,\mathrm{kg}$;
- $r$ = $100\,\mathrm{m}$, so the diameter of the thing is going to be $200\,\mathrm{m}$, which I'm assuming is the largest structure you can plausibly build and protect (see below);
- $v = 1.2\times 10^4\,\mathrm{ms^{-1}}$ (a bit over escape velocity for the Earth: orbital velocity is less of course, but it's not that much less);
- $u = 10^{10}\,\mathrm{Pa}$, which is perhaps plausible.
So this gives
$$d \ge 1.35\,\mathrm{cm}$$
So, well you could probably build such a thing, but I'm pretty sure the 'light cable' assumption is wrong and you'd have to take account of the mass of the cable. This might kill you, but my intuition is it won't.
One additional thing we can work out (thanks to Christopher James Huff for pointing out that I probably should) is what the centripetal acceleration of the thing is just before launch. From the expressions $v = r\omega$ and $a = r\omega^2$ it's easy to get $a$ in terms of $v$ and $r$:
$$a = \frac{v^2}{r}$$
This shows why larger structures are better, but also why higher launch velocities are bad news. For our proposed $100\,\mathrm{m}$ radius launcher, at escape velocity, we get $a \approx 1.4\times 10^6\,\mathrm{ms^{-2}} \approx 145000\,g$, where $g$ is the acceleration due to gravity. The object we're launching is going to have to be very, very tough.
Notes
Things get better the larger you make the structure, because the acceleration goes down as it gets bigger. But I think there are practical limits to how big you can make the structure. In particular if the cable breaks just before launch then the objects you are about to launch will hit the structure at roughly escape velocity. For my $1\,\mathrm{kg}$ masses the energy you need to absorb is $1.4\times 10^8\,\mathrm{J}$, which is the equivalent of about $34\,\mathrm{kg}$ of TNT. And you probably want to launch substantially more than that mass.
Indeed, when you let go the mass you want to launch then you have to deal with the other mass anyway. If you want to launch a tonne, then you have to deal with something equivalent to exploding $17\times 10^3\,\mathrm{kg}$ of TNT. This is equivalent to a large conventional bomb (an earlier version of this answer compared it to the Trinity test because I got kilogrammes & tonnes confused when thinking about it: it's nowhere near that).
This is why I assume you can't build a really large structure: if you want to launch a significant mass then you need to deal with something equivalent to the explosion of a nuclear weapon happening inside the structure, anywhere. This has to be a really substantial structure, and building a really large one will be very, very expensive.
Note that this sort of thing is a problem for any kinetic-energy launch system: if you are going to launch a mass $m$ at velocity $v$ then it's going to have energy of $mv^2/2$ at the point of launch, and you need be ready to dissipate that energy if it is released really abruptly. Of course a rocket-based system also has to deal with dissipating all the energy stored in the fuel, but fuel explosions are a lot less abrupt than something hitting you, and they also have the advantage that the object causing the trouble is moving relatively slowly so you can reliably predict where the trouble will be.
Why I think the whole idea is silly in practice
Quite apart from the fact that doing anything serious with this involves containing explosions equivalent to nuclear weapons and building payloads which can withstand tens or hundreds of thousands of gravities of acceleration, there is a question of what happens to the object you have launched. In particular this object is travelling at escape velocity through dense atmosphere. I'm not competent to do the sums, but I imagine that this is just catastrophic: how much energy does it lose? How much faster do you have to launch is as a result, naking everyting even worse? How hot does it get, & what do you have to make it out of to ensure it can survive. What happens to anything near the launch site?
I think it's all just mad: this whole idea is a silly toy.
