No. They could have went on forever having not eaten of the forbidden tree. Their immortality would remain intact so long as they left it alone. Nothing is implied that they must disobey and eat of the tree to become mortal beings, falling from imortality.
In fact, A&E never had immortality and so did not lose it.
We agree that they were never immortal so your statement "Their immortality would remain intact so long as they left it alone" is simply nonsense. They were NOT immortal.
They were dumb ignorant animals. Saying "Nothing is implied that they must disobey and eat of the tree to become mortal beings, falling from imortality" is doubly wrong. First forbidding them from eating from the Tree of the Knowledge of Good and Evil would have NO meaning. They don't know right from wrong. It is like telling a snail not to climb the wall. It has no knowledge of right or wrong and so no way to obey. Second, they could not fall from something they never had.
Finally, Adam and Eve were NOT thrown out of the Garden of Eden because they ate from the Tree of Knowledge, but because God feared they would eat from the Tree of Life.
Aslan is not a
Tame Lion