Calabi-Yau theorem implies that any diffeomorphism of a Calabi-Yau manifold which preserves
the complex structure and the Kahler class also preserves the Calabi-Yau metric. However, the group of isometries of a compact metric space is compact. The group of isometries of a Riemannian manifold
is a Lie group by Meyers-Steenrod. To prove that the group of polarized isometries of K3 is finite, it remains to show that it is 0-dimensional. This would follow if we prove that there are no holomorphic vector fields. However, K3 is symplectic, hence holomorphic vector fields are the same as holomorphic 1-forms, and they all vanish because $b_1(M)=0$.
For an algebro-geometric argument, please see the comment of abx above this.