What happens when P controller's gain equals zero?
There will be no input to the plant, if \$K = K_p = 0\$ (and \$K_i = 0, K_d = 0)\$.
The controller is ignored in this case (not if \$K = 1\$, because you still feedback the output).



(Original picture taken from Wikipedia and modified)

With \$K = 0\$ the system is stable, because the equilibriums of your plant are stable.



Expanding based on your comment: The block diagram and the transfer funtions are just models for the system you want to describe. So for \$K=0\$ this could mean, that the controller is connected, but no control signal is applied to the plant (because you multiply by 0). Setting \$K=0\$ could also mean that the controller is not connected at all.
For example you want to simulate what happens, when the P-Controller fails. You could do this by setting \$K=0\$. 

For \$K=1\$ the controller is not ignored. As long as you got the feedback you respond to changes in the output. Lets say \$v(t)\$ is the input to the plant.
$$v(t) = K_p \cdot e(t) = K_p \cdot (u(t) - y(t))$$
For \$K=1\$ you are still feeding back \$y(t)\$.
$$v(t) = e(t) = u(t) - y(t)$$
It would be wrong to say, that the controller wouldn't do anything. It just doesn't amplify the signal.