(1) Background: Maintaining soil fertility and crop productivity using natural microbial diversity could be a feasible approach for achieving sustainable development in agriculture. In this study, we compared soils from vineyards under organic and conventional management by predicting functional profiles through metagenomic analysis based on the 16S rRNA gene. (2) Methods: The structure, diversity and predictive functions of soil bacteria related to the biogeochemical cycle of the soil were analyzed, including oxidative and hydrolytic C-cycling enzymes, N-cycling enzymes and P-cycling enzymes. The inter-row spontaneous vegetation in the organic vineyards was also characterized. (3) Results: A clear effect of the farming system (organic vs. conventional) and cover management (herbicides plus tillage, mowing only and mowing plus tillage) on bacterial beta diversity and predicted functions was evidenced. While conventional viticulture increased the potential capacity of the soil to regulate the cycling of inorganic forms of N, organic viticulture in general enhanced those functions involving organic N, P and C substrates. Although the soil bacterial community responded differently to contrasting soil management strategies, nutrient cycling and carbon sequestration functions remained preserved, suggesting a high bacterial functional redundancy in the soil in any case. However, most of the predicted bacterial functions related to soil organic matter turnover were enhanced by organic management. (4) Conclusions: We posit the potential for organic viticulture to adequately address climate change adaptation in the context of sustainable agriculture.