folks, I'm new here, but not to the subject of Edgar Rice Burroughs and his characters, If I might offer my opinions...
The term "Immortal" is not interchangeable with the term "invulnerable." Yes, John Carter is immortal (Burroughs at one point refers to him as "the unaging Virginian) He cannot grow old, or, at least, he doesn't age as normal humans do. He can, however, be killed just like any other human. As Burroughs wrote him, however, as the greatest swordsman who ever lived, living in a society where single combat with swords is the highly preferred weapon of choice, him being killed is a highly unlikely event.
Carter's extended lifespan works out quite well on Mars, where the inhabitants live for as much as a thousand years, with those lucky enough to survive that long in Mars' warlike civilization show almost no signs of aging until the very last few years. If he were a normal mortal, he would grow old and die long before his beloved Dejah Thoris showed even the slightest signs of aging.
Incidentally, I might add that, by one method or another, all of Burroughs' series focus on a character who is unaging. Besides Carter, there is Carson Napier on Venus, who doesn't age because of an elixir given to him by the Venusians, Tarzan, who was granted immortality as a reward for saving a witch doctor, and David Innes, who explains his lack of aging by saying that, since time does not seem to exist in Pelucidar, his body simply doesn't realize that it should be growing older, therefore it remains young.