The situation you are describing is well-studied: the stress distribution is known as Hertzian Stress.
Because the substrate (the surface that the sphere is resting on) and the sphere itself both have a finite elastic modulus, the surfaces will deform and the actual contact area will be a circle of finite size; the stress distribution (as a function of radius) will follow a quadratic law (as a function of distance from the center of the contact patch).
From the link above, a few key results. First, the radius $a$ of the contact area for a sphere (radius R, elastic modulus $E_1$, Poisson ratio $\nu_1$) on a plane with elastic modulus $E_2$ and Poisson ratio $\nu_2$ when subject to a force $F$ (which may be just the weight of the sphere) is given by
$$a = \sqrt{\frac34 FR\left(\frac{1-\nu_1^2}{E_1}+\frac{1-\nu_2^2}{E_2}{}\right)}$$$$a = \sqrt[3]{\frac34 FR\left(\frac{1-\nu_1^2}{E_1}+\frac{1-\nu_2^2}{E_2}{}\right)}$$
And the maximum pressure is
$$P_0 = \frac{3F}{2\pi a^2}$$
The distribution of pressure with radial distance is then given by
$$P(r) = P_0 \left(1-\frac{r^2}{a^2}\right)$$$$P(r) = P_0 \sqrt{1-\frac{r^2}{a^2}}$$
as given here
Edit: Fixed formula.
