Can your body fight and get rid of pneumonia on its own?

Maybe, maybe not. Since some are triggered by viruses, it is certainly possible. However, the creator of the "Muppets" died rather suddenly of a treatable pneumococcal pneumonia that he ignored in its earliest stages. Makes one consider the possibilities.
Maybe. It is possible, not everyone died of pneumonia in the pre antibiotic era, but it is not wise to do this if you have a choice. Anyone who thinks they may have pneumonia should seek prompt medical attention to avoid serious complications.