I think you can, grammatically speaking anyway. Take the case of "I regret to inform you that the 8.36 to Little Snotsbury is cancelled due to leaves on the track' - no-one supposes the train announcer is personally responsible or in control of the leaves, or is actually apologising for the act of announcing the cancellation which is (a) her job and (b) important and useful information for the passengers. You might say it's a very trite phrase as the announcer doesn't give a shit about the leaves or the passengers but suppose she is in fact a sincere and empathetic announcer, she feels truly sorry that the commuters are going to be late for work, she's ashamed her useless rail company bosses haven't sorted the leaves, she wishes she had the power to make things better, I'd say that's 'regrets' even though obviously she has no actual power over the events... Maybe not huge, life altering, worthy of a deathbed mention type of regrets but to me regrets none the less.
Talking emotionally/psychologically I'd say the sort of regrets you feel about things you have done personally or in control of though are much more powerful than the ones where the situation was entirely outside your control but you wish it could have been different. I certainly have both kinds in my life, although the greatest sadnesses 'big ticket regrets' around deaths, serious illnesses etc are of course things outside my control, the things I keep coming back to are the more personal regrets, I wish I'd been kinder to that person, I wish I'd spent more time with my loved one while I had the chance, I wish I hadn't wasted so much time and energy worrying about things that never happened, or pursuing goals I was never able to achieve, that kind of thing, things absolutely within my own control (recognizing we all are the products of our environment and social structures, how much agency do we really have yadda yadda). That's what I would mean by saying of course I have regrets. Not to say I'd ultimately change my life now if I had the chance, especially if it would mean not having my DC or DH, but I still wish I could change those things and have my life now, but take someone else's point on board that without the learning and personal development from those mistakes I wouldn't be the person I am now...