Is there a formalization of normalization of impredicative system F? In particular Agda seems not strong enough to prove that.

Is the predicative Calculus of Inductive Constructions universes (Coq without Prop) sufficient?

How about with the impredicative Prop?
Coq without Prop is not strong enough, because it's basically Martin-Löf type theory with universes.
Coq with Prop is strong enough, because you can encode sets of normalizing terms via predicates $S : \mathrm{Term} \to \mathrm{Prop}$, and impredicative universal quantification lets you express arbitrary intersections.