Pandemic influenza A viruses can emerge from swine, an intermediate host that supports adaptation of human-preferred receptor-binding specificity by the hemagglutinin (HA) surface antigen. Other HA traits necessary for pandemic potential are poorly understood. For swine influenza viruses isolated in 2009–2016, gamma-clade viruses had less stable HA proteins (activation pH 5.5–5.9) than pandemic clade (pH 5.0–5.5). Gamma-clade viruses replicated to higher levels in mammalian cells than pandemic clade. In ferrets, a model for human adaptation, a relatively stable HA protein (pH 5.5–5.6) was necessary for efficient replication and airborne transmission. The overall airborne transmission frequency in ferrets for four isolates tested was 42%, and isolate G15 airborne transmitted 100% after selection of a variant with a stabilized HA. The results suggest swine influenza viruses containing both a stabilized HA and alpha-2,6 receptor binding in tandem pose greater pandemic risk. Increasing evidence supports adding HA stability to pre-pandemic risk assessment algorithms.