Arwen was able to choose to give up her Immortality because she was a Half-elf, and they were given a choice as to which fate would be their's. This was indeed because her father was a Half-elf, since his parents were Eärendil and Elwing who were both Half-elven. (Eärendil's father, Tuor, was a Man, and his mother, Idril, was a Noldorin Elf)
Would all Elves have become Mortal? No, I don't think that's possible, since death was Eru's gift to Mortals (Men, Dwarves, etc.) and the Elves could never really have it. Even when they died, it wasn't the same as when a Mortal did. The exception is of course Lúthien, who managed to convince Ilúvatar Himself to let her live another life with Beren as a Mortal.
Hope that made sense, and I hope I understood what you were saying.
