In robotic applications, a key requirement for safe and efficient motion planning is the ability to map obstacle-free space in unknown, cluttered 3D environments. However, commodity-grade RGB-D cameras commonly used for sensing fail to register valid depth values on shiny, glossy, bright, or distant surfaces, leading to missing data in the map. To address this issue, we propose a framework leveraging probabilistic depth completion as an additional input for spatial mapping. We introduce a deep learning architecture providing uncertainty estimates for the depth completion of RGB-D images. Our pipeline exploits the inferred missing depth values and depth uncertainty to complement raw depth images and improve the speed and quality of free space mapping. Evaluations on synthetic data show that our approach maps significantly more correct free space with relatively low error when compared against using raw data alone in different indoor environments; thereby producing more complete maps that can be directly used for robotic navigation tasks. The performance of our framework is validated using real-world data.