If you take “causal determinism” to mean that, for a theory, if you are given an complete set of data about an arbitrary isolated system/the Universe at some arbitrary initial time, then you can completely determine its state at any other - then yes the many-worlds interpretation (MWI) fits this bill.
MWI specifically states:
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the universal wavefunction is objectively real (up to gauge symmetry, presumably) and contains all the data about the Universe.
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the only time evolution that the universal wavefunction undergoes is deterministic and unitary.
So by asserting the reality of the Universal wavefunction and specifying that it evolves in time in such a way that its state at a given time can be determined from its state at any other time, MWI is “causally deterministic”.
However there are some problems of how the universal wavefunction relates to the objective reality we see. Even if we were some supreme being with complete knowledge of the universal wavefunction it is not clear how we could, or even if we would, be able to predict the observations made by us mere mortals. So it could be argued that the causal determinism of MWI is hidden in the same manner as a hidden variables theory and worse you may still need to introduce probability to relate it to the world as we actually observe it.
As an aside, any theory/interpretation which has probabilistic behaviour as fundamental cannot meet the above definition of “causal determinism”, however being non-probabilistic is a necessary, rather than sufficient condition of “causal determinism”. For example general relativity does not have any fundamental probabilistic elements to it, but fails to meet the definition as stated above. That said, the condition is often imposed in GR at the expense of solutions that allow things like time travel or the ability to open a wormhole (for example).