Melt ponding on Arctic sea ice is a key indicator of the transition from a predominantly perennial to a seasonal sea-ice cover, yet quantitative data on pond depth remain limited. Here, we present the first analysis of melt-pond depth using Ice, Cloud, and land Elevation Satellite-2 (ICESat-2)'s Advanced Topographic Lidar Altimeter System (ATLAS). The Density-Dimension Algorithm for bifurcating sea-ice reflectors (DDA-bifurcate-seaice) automatically detects multiple surface returns in ICESat-2 photon data and estimates corresponding surface heights, enabling melt-pond-depth retrievals under varied noise conditions. Airborne lidar and imagery collected during the NASA ICESat-2 Project Arctic Summer Sea Ice Campaign (July 2022) provide near-coincident observations used to evaluate and optimize the algorithm's melt-pond detection. Evaluation of the melt-pond-depth quantile using Chiroptera data shows that the uniform value used in the ATL07 release 7 data product is near-optimal. We demonstrate DDA-bifurcate-seaice's capability to detect a wide range of melt feature morphologies, including smooth or rough bottoms, ridge-adjacent ponds, partial drainage and seawater intrusion. To further improve depth determination, we propose a depth-quantile function that reduces bias and mean-squared error by a factor of 2.75 and 2.2, respectively. This work improves melt-pond-depth estimation using the DDA-seaice-bifurcate, supporting Arctic- and Antarctic-wide mapping in the ICESat-2/ATLAS experimental sea-ice melt-pond data product on ATL07 (release 7).