Will two spherical bodies meet at their center of mass of only gravitational force exists between them?

Two spherical bodies of different radii and different masses are separated by a certain distance in free space.

• If they attract each other only due to gravitational force only, will they meet at their center of mass ?

• Why or why not?

• What would be the relevant equations to find their final meeting position (along the line joining their centers) ?

Intuitively it feels that they should meet at their center of mass, however I am not able to mathematically verify it.

• Think about it. If you have two billiard balls, can their centers of mass really coincide? Commented Mar 1, 2017 at 12:10
• @ChesterMiller Yes, I understood. However, how to calculate the meeting point of the two spheres (where they touch) ? Commented Mar 1, 2017 at 12:49

For example, take two spherical masses, each of mass 1 kg. Let their centers lie at $(-1,0)$ and $(1,0)$. Obviously, the center of mass of the system is at $(0,0)$.
Now what if one of the spheres had a radius of $1.5$ units and the other a radius of $0.5$ units?