Sea ice and climate models are key to understand and predict ongoing changes in the Arctic climate system, particularly sharp reductions in sea ice area and volume. There are, however, uncertainties arising from multiple sources, including parametric uncertainty, which affect model output. The Los Alamos Sea Ice Model (CICE) includes complex parameterizations of sea ice processes with a large number of parameters for which accurate values are still not well established. To enhance the credibility of sea ice predictions, it is necessary to understand the sensitivity of model results to uncertainties in input parameters. In this work we conduct a variance-based global sensitivity analysis of sea ice extent, area, and volume. This approach allows full exploration of our 40-dimensional parametric space, and the model sensitivity is quantified in terms of main and total effects indices. The global sensitivity analysis does not require assumptions of additivity or linearity, implicit in the most commonly used one-at-a-time sensitivity analyses. A Gaussian process emulator of the sea ice model is built and then used to generate the large number of samples necessary to calculate the sensitivity indices, at a much lower computational cost than using the full model. The sensitivity indices are used to rank the most important model parameters affecting Arctic sea ice extent, area, and volume. The most important parameters contributing to the model variance include snow conductivity and grain size, and the time-scale for drainage of melt ponds. Other important parameters include the thickness of the ice radiative scattering layer, ice density, and the ice-ocean drag coefficient. We discuss physical processes that explain variations in simulated sea ice variables in terms of the first order parameter effects and the most important interactions among them.