This paper presents a novel characterization of completely passive linear quantum stochastic systems solely in terms of the controllability Gramian of such systems. Importantly, exploiting this result, we establish a connection between the class of completely passive systems and the class of linear quantum stochastic systems with a pure Gaussian steady state. Consequently, we obtain a new complete parameterization for the class of systems generating a pure Gaussian steady state. Examples are provided to illustrate our results.