The answer is that the Prophet, peace and blessings be upon him, did not explicitly state that Ali was the caliph after him. Rather, he said, “You stand to me as Aaron stood to Moses,” when he left him in charge of his family, just as Moses left Aaron in charge of his people during his lifetime. This is akin to deputizing someone or appointing an agent. It is as though a man were to travel to another land and say to someone else, “Act on my behalf with regard to my family and take care of their needs.” This does not mean that he is his appointed successor after his death; rather, it indicates that he is his deputy during his lifetime for a limited period.
The caliphate, on the other hand, takes place only after the death of the first caliph and his final departure from this world. He cannot dismiss the caliph who succeeds him, because he is dead. This case, however, differs: he can dismiss him by sending word to him and saying, “I have terminated your appointment over my family.”
Once Abu Bakr’s caliphate is established, Umar’s caliphate is established; and once Umar’s caliphate is established, Uthman’s caliphate is established, may Allah be pleased with him. This is why the Rafidah do not acknowledge Abu Bakr’s caliphate: so that they may dismantle his caliphate, and that of Umar and Uthman, and arrive at a caliph who was not appointed as successor—a caliph who himself, may Allah be pleased with him, pledged allegiance to Abu Bakr and Umar, may Allah be pleased with them. He would say, may Allah be pleased with him, “The best of this community after its Prophet is Abu Bakr, then Umar.” (1)
Thus, the caliphate is established by textual designation—that is, when the first caliph designates so-and-so as the caliph after him.