Petrified wood is a fossil. Yes, it is a natural process. I suppose you could recreate the process in a lab setting, which would make it not natural. Because petrified wood is a fossil, the natural process to make it is not any different than making any other kind of fossil.
In the case of petrified wood, start with some kind of plant material. Let's use a woody stem. Bury it. Perhaps a landslide buried it. Because of its rather swift encasement, it could be protected from decaying. Ground water that is rich in dissolved solids will flow through the sediment and replace the woody stem with silica, pyrite, calcite, or other inorganic matter. What results is a fossil of the original plant part.