Is rotational motion of the centre of mass impossible? We know that for a system, the center of mass $CM$ moves as a particle as though all the forces on the system were acting on it. So does that mean rotational motion of the center of gravity impossible? 
 A: Firstly, an object rotates about an axis (or two or even three, in which case these rotational modes are independent of each other), not a point. These axis do not have to run through the centre of gravity (CoG) of the object: see the Earth's rotation around the Sun, e.g.
For argument's sake let's take a simple case of a spinning top rotating about an axis that does run through its CoG. Assume also no external forces or fields act on it at $t \leq 0$ and that the object was motionless.
Newton now tells us (analogously to $F=ma$) that:
$$\tau=I\alpha,$$
where $\tau$ is a torque (moment) applied to the object, $I$ the moment of inertia of the object about the axis of rotation and $\alpha$ the angular acceleration.
Also:
$$\alpha=\frac{d\omega}{dt},$$ 
with $\omega$ the angular speed.
If at $t=0$ we apply this torque for an interval of time $\Delta t$, then:
$$\omega=\alpha \Delta t=\frac{\tau}{I}\Delta t$$
The object has acquired angular speed and is now rotating. At $t=\Delta t$ we withdraw the torque $\tau$. Newton's Laws of motion now tell us that the state of motion of the object will remain unaltered because no external forces or fields act on it: it will basically continue to rotate at the same constant speed 'forever'.
To stop or to decelerate the existing rotation we would have to apply another torque, say $\tau'$ with opposite sense of $\tau$.
A: If there is no external field acting on the system, yes. No rotational motion (in the sense that the center of mass rotates around a point in space which is different from its position) is allowed due to the law of conservation of momentum. Should there be an external field acting on the system, the conservation of momentum is violated (in the sense that we do not take into account the variation in the momentum of the fields, and thus it seems violated) and the centre of mass can rotate.
Consider as an example two electrons orbiting around a proton. The centre of mass of the two electrons obviously rotates around the proton together with the electrons. This is because we can consider the electromagnetic field produced by the proton as an "external" field, if we are not willing to go through the description of the dynamics of the proton. On the other hand, the physically complete description of the system should also include the dynamics of the proton. In this description, you have to focus on the centre of mass of the three particles, and this centre of mass (assuming it is not subject to yet some other kind of external force) does not rotate around any fixed point in space.
