To remove vulgarity, one must first define vulgarity.
In a case with a famous outcome the Justice Stewart wrote, "I shall not today attempt further to define the kinds of material I understand to be embraced within that shorthand description; and perhaps I could never succeed in intelligibly doing so. But I know it when I see it, and the motion picture involved in this case is not that."
In order to not explicitly attempt to remove what someone may or may not consider vulgar, usage of "abridge" might be suitable.
The disclaimer, "This film has been modified from its original version. It has been formatted to fit this screen and edited for content." is used when showing a movie for TV (often for basic cable or broadcast).
So, while "abridged" and "edited" are not as fancy as "bowdlerize", these gain an advantage. If the recipient finds something that they consider to be vulgar, they can't state that the editor failed to remove the vulgarity since this varies from person to person.