The rule is indeed the same as the English “saw”.

J’ai vu le film

and

J’ai regardé le film

are both valid ways of saying “I saw/watched the movie”. I personally don’t find any difference in meaning between the two, though I guess you would write the one with “regarder” rather than “voir” in text.

Note however that this does not necessarily apply to similar things. For example, you could say

J’ai regardé la télévision

or

J’ai regardé la chaîne France 2

but not

J’ai vu la télévision (unless you’re talking about seing the object TV)

or

J’ai vu France 2

This is also true in English where you wouldn’t say

I saw CNN

This does not really explain why both “to see” and “voir” have that same accepted use for movies.