I think "I saw both movies" is closer to «Я ба́чив оби́два фі́льми» in meaning.
Imho "I (have) watched both movies" is much closer to "Я дивився обидва фільми".
Google Translate says бачив = saw and дивився = watched.
Okay, so we're dealing with two different verbs here: one meaning "to see," and one meaning "to watch." Fair enough. But what is going on with the conjugation? I understand that "бачив" is in the past tense because it ends with "-в," but why does "дивитися" end in a "-я" when conjugated?
EDIT: It seems "дивився" is conjugated in the past tense ("дивив-") with the reflexive suffix added ("-ся").
You are right.
Я (по)дивився... -> I (have) watched
Я бачив -> I saw, I have seen
But in case of movies I think it's pretty common to say "Oh, I've seen that movie!" in English (rather than "watched"), and in Ukrainian we'd say either "О, я бачила цей фільм!" or "О, я дивилася цей фільм!". So in this sentence I would allow interchanging see/watch and дивитися/бачити.