I think "I saw both movies" is closer to «Я ба́чив оби́два фі́льми» in meaning.
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.