Is "assume" an appropriate verb to use in this sentence?
We all need to acknowledge the great importance science is assuming nowadays.
Yes, this is a perfectly appropriate and natural use of "assuming*.
See sense C2 in the Cambridge definition:
to take or begin to have responsibility or control, sometimes without the right to do so, or to begin to have a characteristic:
- The new president assumes office at midnight tonight.
- The terrorists assumed control of the plane and forced it to land in the desert.
> * The issue has assumed considerable political proportions (= has become a big political problem).