The sensitivity and uncertainty analysis tools of the Oak Ridge National Laboratory SCALE nuclear modeling and simulation code system that have been developed over the last decade have proven indispensable for numerous application and design studies for nuclear criticality safety and reactor physics. SCALE contains tools for analyzing the uncertainty in the eigenvalue of critical systems with realistic three-dimensional Monte Carlo simulations but currently can only quantify the uncertainty in important neutronic parameters such as multigroup cross sections, fuel fission rates, activation rates, and neutron fluence rates with one- or two-dimensional models. A more complete understanding of the sources of uncertainty in these design-limiting parameters using high-fidelity models could lead to improvements in process optimization and reactor safety and help inform regulators when setting operational safety margins. A novel approach for calculating eigenvalue sensitivity coefficients, known as the CLUTCH (Contributon-Linked eigenvalue sensitivity/Uncertainty estimation via Track length importance CHaracterization) method, was recently explored as academic research and has been found to accurately and rapidly calculate sensitivity coefficients in criticality safety applications. The work presented here describes an extension of the CLUTCH method, known as the GEneralized Adjoint Responses in Monte Carlo (GEARMC) method, that enables the calculation of sensitivity coefficients and uncertainty analysis for a generalized set of neutronic responses using high-fidelity continuous-energy Monte Carlo calculations. Several criticality safety systems were examined to demonstrate proof of principle for the GEAR-MC method, and GEAR-MC produced response sensitivity coefficients that agreed well with reference direct perturbation sensitivity coefficients.