We already had some threads about this question. Some people speculated, that Alexander would conquer Arabia next, as perhaps already planned. And from there go to Africa and Italy.
I doubt, Alexander would even conquer Arabia. He would be immediately busy to really administrate this huge empire he had established provisorily, and fight with all the issues, intrigues, local riots and usurpations. The real hard work for an emperor started exactly now, when he was back to Babylon. As the history of the Seleucids, the Asarcids and the Sassanids shows clearly. And I doubt, Alexander was willing to face this challenge and commit himself to this hard job.
If Alexander ignores the need to actually rule his empire and prefers to hurl himself into the next adventure, his empire would break into pieces sooner or later and he is enforced to go back and deal with it. Hence, he would never see Carthago or Rome. Perhaps even not Arabia Felix.
Alexander had the mercy of an early death. He never had to prove, that he is not just a genious and brave general, but also a competent ruler. Even if he proves to be a competent ruler, stays in Babylon and faces the challenge, he and his hopefully also competent descendants would have a very hard time to keep this empire together.