"There is no ground to assume the continuity of causation or motion...without taking for granted that things will be tomorrow as they have been today."

I'd contend that is an epistemological assertion, not a metaphysical one.

Our ability to do physics and metaphysics alike depends on our epistemology.

The Aristotelian distinction worth keeping in mind is physics is the study of that which is real and changes, while metaphysics is the study of that which is real and is unchanging.

With that distinction, it seems clear to me that, while our metaphysics can color our understanding of physics, our ability to study changing things does not wholly depend on whether we have a correct metaphysics. 

 Our epistemology, also, cannot but presuppose a certain metaphysic. 

To channel Van Til, "the unbeliever can count, but he cannot account for counting." This applies to all scientific endeavor. 

 ...and, our ability to do physics relies on an objectively real world and objectively real natural systems and processes--whose persistence itself relies on its Creator and his promise to keep it going that way (at least for a while longer).  

 Agreed.  The distinction I'd make from here is that those foundational epistemological axioms don't necessarily require an *explicit* metaphysics, even if they require an implicit one.

The physicist takes as axiomatic, explicitly or implicitly, that reality is consistent, that it is knowable (at least in part), and that physical matter is a relevant object of study.  Those axioms are, it would seem, sufficient for the scientific method.

Under those axioms, metaphysical truths remain in a numinous realm beyond our ability to study.  One might admit those metaphysical presuppositions are necessary for doing physical science while not admitting them as relevant objects of study.

Thus, like any logical system, science needs axioms to get going, but that is an implicit metaphysics; science may remain agnostic about other metaphysical pursuits. 

 Exactly!  If we assume that we are just a chemical accident, why should we trust our senses or our logic or assume that we can know anything.  If we assume that we are created in the image of God and that God designed everything that is, then it is an obvious conclusion to trust our senses and logic and that the universe works according to certain reliable, trustworthy, continual rules and can be known.