Can you "regret" someone else's action?

I have rarely heard regret used like this, and while it sounds wrong to me, the dictionary doesn't appear to preclude this usage. Dictionary.com:

Regret

1. to feel sorrow or remorse for (an act, fault, disappointment, etc.): He no sooner spoke than he regretted it.

2. to think of with a sense of loss: to regret one's vanished youth.

For example, is the following correct?

He always regretted her impulsive decisions.

Oxford American Dictionary includes the following example of the first definition:

I regretted that he did not see you.

So it seems like it's possible to regret something that you didn't cause. Although this is not the common use, I also can't think of a better word.

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.