The word ball has several meanings. I am not sure that every meaning has the same origin but the word ball as in basketball, football, baseball etc comes from the greek verb βάλλω (vallo -that means shoot). From this verb comes also originates and the word ballista (in Greek Ballistra) the weapon that was shooting rocks shaped as balls.
In Greek the word ball is bala (μπάλα). It is counter-loan from English. If the word was earlier in Greek it would be pronounced vala and not Bala.
Ball
