Chemistry - Why does ozone, being heavier than air, not settle down?

While gravity does cause some separation of atmospheric constituents at high altitudes, the reason why ozone doesn't migrate to the ground has more to do with its chemistry.

Ozone is pretty unstable and is formed from some chemical reactions and from the interaction of oxygen with UV light. But it is unstable and decays fairly quickly. This means that is only appears anywhere when the reaction forming it keeps happening. This is true in the upper atmosphere which is why we have an ozone layer. Some hydrocarbons (not just from pollution from cars but also from volatile stuff released by trees) can react to generate ozone at ground-level. But we only see a significant concentration when the reaction forming it is happening.

The effect of gravity on the separation of the components of the atmosphere is small, especially in the lower atmosphere where diffusion and turbulent circulation mix things up thoroughly. In the upper atmosphere the dominant process is diffusion and gravity can have more of an effect. This paper describes the process and has some details of the calculations about the size of the effect. It summarises the situation like this:

Turbulent mixing keeps the relative concentrations of gases nearly constant in the lowest 100 km. At higher altitudes, molecular diffusion controls the concentrations, with the lighter gases becoming relatively more abundant with increasing altitude.

It is a worthwhile read if you want the detail.