No it can't. There have been many attempts and they don't work. The Born rule is independent of unitary evolution. The closest one can get is to declare that the quantum state is fundamentally a statistical object (i.e. the only information in it is observation probabilities) and then with certain assumptions about the size of the state space you can show that the Born rule is the only possible rule for connecting the state to statistics consistent with the unitary dynamics.

So under the assumption that the state encodes probabilities, state space assumptions and consistency with unitary evolution you get the Born rule. However this is not the same as the Born rule arising dynamically from unitary evolution alone.