We introduce FloeNet, a graph neural network trained to emulate the Geophysical Fluid Dynamics Laboratory global sea ice model, SIS2. FloeNet is a mass-conserving model, emulating 6-hr mass and area budget tendencies related to sea ice and snow-on-sea-ice growth, melt, and advection. We train FloeNet using SIS2 data from a reanalysis-forced ice-ocean simulation and test its ability to generalize to pre-industrial and 1% CO2 forcing conditions. FloeNet outperforms a non-conservative model at reproducing sea ice and snow-on-sea-ice mean state, trends, and inter-annual variability, with volume anomaly correlations above 0.96 in the Antarctic and 0.76 in the Arctic, across all forcings. FloeNet also produces the correct thermodynamic-dynamic response to forcing, enabling physical interpretability of emulator output. Finally, we show that FloeNet outputs high-fidelity coupling-related variables, including ice-surface temperature, ice-to-ocean salt flux, and melting energy fluxes. We hypothesize that FloeNet will improve polar climate processes within existing atmosphere and ocean emulators.