Can you "regret" someone else's action?

I regret his decision, I regret his leaving:

Though it doesn't sound ungrammatical ( at least to me) there is not much evidence of "regret" used with respect to someone else's actions. I'd say that usage of "regret" is mainly with respect to one's action.

As for grammaticality your sentence "He always regretted her impulsive decisions." is entirely acceptable and immediately understandable. (Data point of 1 native BrE speaker here.)

As for its usage, again, yes one can certainly "regret" someone else's actions; only the other day I caught a snippet on the TV about a man in California regretting his son's actions in killing a number of people. I don't recall that he used the word "regret" but that was certainly what he was doing.

Empathy not regret should be the proper term unless there is reason for a guilty conscience. Getting the accused in a legal trial to admit regret over an alleged action will surely conclude negatively for that individual if they are claiming innocence; I posit this shows my position to be falsifiable.

Regret, as commonly understood, always implies a sense of disappointment about one's own decisions or actions. Those uses of regret which seem to refer to the decisions or actions of others are misuses of the word in order to avoid acknowledging one's own resentment of those decisions/actions.

If the intent is truly to convey feeling bad about what someone else did in a non-condescending way or without trying to shift responsibility off of oneself, then the appropriate word is lament.

Late to the discussion here. Such a use does not sound correct to me, but I acknowledge that it is used on rare occasions. I agree with the respondent that such occasions would be more honestly served by “lament”. Often regret about others actions is used by politicians, and may carry an implied threat eg. The US regrets the decision of the United Nations.