Bryan Tan answer has described most of the part. I am going to add some part where they actually fight.
Natsu Dragneel is E.N.D, Etherious Natsu Dragneel, brother of Zeref Dragneel, died 400 years ago and resurrected as Etherious demon but not from the Book of Zeref.
If E.N.D is the strongest demon in the Books of Zeref and if Acnologia can slay countless dragons, is it possible for E.N.D to slay Acnologia?
No, E.N.D(Natsu) can't win Acnologia if they fight one on one. Natsu uses Dragon slayer magic. Acnologia was a dragon slayer too, who turned into a dragon because he overused his dragon slayer magic.
In Chapter 540-44, when all the dragon slayer are trapped in Ravines of time, they all together fight with Acnologia. If we see the fights between all the dragon slayer and Acnologia, he is overwhelmingly powerful.
His dragon body was sealed by Lucy in Fairy sphere with the help of combine power of whole continent which made his body in Ravines of time freeze. Taking this as a chance Natsu defeats Acnologia with the help of the power received from all the dragon slayer.
so is it possible for the Demon to slay a dragon like Acnologia?
No, magic has no effect against him. So slaying Acnologia is impossible.
Somebody might be thinking that if magic can't kill then how Dragon Slayer magic can? Read Dragon Slayer Magic thoroughly for better grasp