The fact that European Union (EU) countries have adopted an ambitious plan to achieve zero greenhouse gas emissions by 2050 requires decisive action within the scope of innovation and of the level of energy consumption, especially of the energy from renewable sources. Being directed toward innovation within the scope of renewable energy technology, as well as the proper management of renewable energy consumption, are the main actions aimed at increasing the efficiency of using clean energy, and which also bring the EU closer to the implementation of the assumptions adopted in the European Green Deal. The aim of our study was to assess the progress toward the management of renewable energy consumption in the innovativeness context and the relationship between energy consumption and selected indicators of innovativeness in European Union countries. We present an original ranking assessment of the progress toward the management of renewable energy consumption and identify relationships between the energy consumption of selected energy sources (both renewable and non-renewable) and of selected innovation assessment indicators. The data used to develop the original rating were optimized using the procedures of the MULTIMOORA method, while the relationships between variables were identified through correlation analysis. Our findings provide evidence of significant relationships between the consumption of selected energy sources (in the group of non-renewable sources, e.g., peat and peat products and oil and petroleum products, and in the group of renewable sources, e.g., wind, biofuels, and renewable waste) and of selected indicators of innovation evaluation (e.g., human resources, finance, and support).