No, it's because John has been a martian all along. Hense, his long life, natural telepathic ability, and ability to interbreed with the egg laying martians.
John Carter was not physically transported to Mars. He "dies" on Earth and then finds himself there. If I recall correctly, this is deemed to be astral travel in the books. So, he's not in his original physical body while on Mars - his ability to interbreed then does not depend on normal biology.
That being said, he was nigh immortal, and could not remember ever having a childhood - so John Carter was no normal human.