We don't know.

Quantum physics seems to suggest that all interactions are probabilistic. If that's strictly true, and the outcomes truly are random (seems to be) then no, things would definitely end up progressing differently.

But the apparent randomness of quantum physics doesn't sit well with people, so there's plenty of theories for how various unobservable mechanisms might restore causality. If one of those were true then yes, things would progress exactly the same from the same initial conditions.