An advanced algorithm for atmospheric water vapor column (WVC) retrieval from the Advanced Microwave Scanning Radiometer (AMSR) measurements over the Arctic sea ice (SI) and open ocean waters is presented. The algorithm is built on the physical modeling of the brightness temperature (BT) of the microwave radiation of the SI-open ocean-atmosphere system at the AMSR frequencies and polarizations. The BTs are calculated using a data set of the SI, atmospheric, and oceanic parameters changing in the range of their natural variability in the Arctic, and using the SI microwave emission coefficients varied according to the published experimental data. The inverse operator explores neural networks (NNs), trained on an ensemble of modeled BTs. The algorithm is applied both to the AMSR-E and to the AMSR2 measurement data. Validation of the algorithm is performed with radiosonde (r/s) WVC measurements from the four Arctic coastal stations at different SI conditions during 2014-2017. The results of the application of the new algorithm to satellite radiometer measurements are also compared with the Era-Interim reanalysis WVC, as well as with other satellite WVC products, based on the data of the Moderate Resolution Imaging Spectrometer (MODIS) and on the data of the Advanced Microwave Sounding Unit-B (AMSU-B) for 2008 and 2015. To justify the usage of the Era-Interim WVC as a reference data set for the algorithm accuracy estimation in the Arctic area, Era-Interim WVC is also compared with the r/s WVC measurements.