Why don't these exceptional languages salute with words anent well-being? Surely their speakers must still care about health and peace! Or do they truly disregard well-being?
Salute goes back ultimately to the Latin noun salūs, a relative of salvus ‘safe, healthy’ (source of English safe and save). This had two main strands of meaning. The primary one was ‘health, well-being’, and in that sense it lies behind English salubrious  and salutary . But by extension it also denoted a ‘wish for someone’s well-being’, hence a ‘greeting’, and it is this that has given English, via its derived verb salūtāre ‘greet’, salute.
Word Origins (2005 2e) by John Ayto. p 434 Right column.