"how can I calculate the maximum allowed contact stress from the material specifications?"
You can't.
You must also characterise the loading conditions, factor of safety required, number of load cycles, mechanical properties of the mating part (MoE), operating environment, etc.
The heat treatment of the material is also very important, if it is a metal. The limiting factor in contact is usually subsurface shear. If you have a case hardened material, you will need to evaluate what depth the max subsurface shear occurs at vs. what properties you have at that depth of the case.