The center of gravity (more precisely, the center of mass) of a system of point masses with masses $m_i$ and distances $\vec{r}_i$ from the origin can be viewed as a weighted mean. It can generally be calculated as follows:
\begin{equation}
{\vec{r}_\mathrm{S} = \frac{\sum_i m_i \vec{r}_i}{\sum_i m_i}}
\end{equation}
If we consider only two masses $m_1$ and $m_2$ and differentiate the resulting equation with respect to time, we obtain
\begin{equation}
\vec{v}_\mathrm{S} = \frac{m_1\vec{v}_1+m_2\vec{v}_2}{m_1+m_2}
\end{equation}
and thus the velocity of the center of mass of the two masses moving at $\vec{v}_1$ and $\vec{v}_2$. This corresponds exactly to the motion after an inelastic collision and is therefore identical to the equation we obtained with the inelastic collision.