Single image surface normal estimation and depth estimation are closely related problems as the former can be calculated from the latter. However, the surface normals computed from the output of depth estimation methods are significantly less accurate than the surface normals directly estimated by networks. To reduce such discrepancy, we introduce a novel framework that uses surface normal and its uncertainty to recurrently refine the predicted depth-map. The depth of each pixel can be propagated to a query pixel, using the predicted surface normal as guidance. We thus formulate depth refinement as a classification of choosing the neighboring pixel to propagate from. Then, by propagating to sub-pixel points, we upsample the refined, low-resolution output. The proposed method shows state-of-the-art performance on NYUv2 and iBims-1 - both in terms of depth and normal. Our refinement module can also be attached to the existing depth estimation methods to improve their accuracy. We also show that our framework, only trained for depth estimation, can also be used for depth completion. The code is available at https://github.com/baegwangbin/IronDepth.