With the birth of the next-generation GPS III constellation and the upcoming launch of the Navigation Technology Satellite-3 (NTS-3) testing platform to explore future technologies for GPS, we are indeed entering a new era of satellite navigation. Correspondingly, it is time to revisit the design methods of the GPS spreading code families. In this work, we develop a natural evolution strategy (NES) machine learning algorithm with a Gaussian proposal distribution which constructs high-quality families of spreading code sequences. We minimize the maximum between the mean-squared auto-correlation and the mean-squared cross-correlation and demonstrate the ability of our algorithm to achieve better performance than well-chosen families of equal-length Gold codes and Weil codes, for sequences of up to length-1023 and length-1031 bits and family sizes of up to 31 codes. Furthermore, we compare our algorithm with an analogous genetic algorithm implementation assigned the same code evaluation metric. To the best of the authors' knowledge, this is the first work to explore using a machine learning approach for designing navigation spreading code sequences.