SARS-CoV-2 has demonstrated a robust ability to adapt in response to environmental pressures---increasing viral transmission and evading immune surveillance by mutating its molecular machinery. While viral sequencing has allowed for the early detection of emerging variants, methods to predict mutations before they occur remain limited. This work presents SpikeGPT2, a deep generative model based on ProtGPT2 and fine-tuned on SARS-CoV-2 spike (S) protein sequences deposited in the NIH Data Hub before May 2021. SpikeGPT2 achieved 88.8% next-residue prediction accuracy and successfully predicted amino acid substitutions found only in a held-out set of spike sequences deposited on or after May 2021, to which SpikeGPT2 was never exposed. When compared to several other methods, SpikeGPT2 achieved the best performance in predicting such future mutations. SpikeGPT2 also predicted several novel variants not present in the NIH SARS-CoV-2 Data Hub. A binding affinity analysis of all 54 generated substitutions identified 5 (N439A, N440G, K458T, L492I, and N501Y) as predicted to simultaneously increase S/ACE2 affinity, and decrease S/tixagevimab+cilgavimab affinity. Of these, N501Y has already been well-described to increase transmissibility of SARS-CoV-2. These findings indicate that SpikeGPT2 and other similar models may be employed to identify high-risk future variants before viral spread has occurred.