Do you think that if you didn't offer free doughnuts this situation probably wouldn't happen?
I guess what I'm saying is that you can't absolve yourself of all responsibility for this child or other children getting fat.
I think that this is something the company offers, and it's part of OP's job to hand them out. It's not OP's idea and, even if it was, the issue of taking responsibility for your own health is still in question. S/he is not forcing anyone to take anything they don't want, just because it is free.
Same as those people handing out free samples in a supermarket don't corner you and force feed you the samples. If you don't want it, don't take it.
Also, if you had offered the doughnut to the child and the mother had said no (ie. she was acting responsibly regarding her child's health) this might have upset the kid and put pressure on the mother to give in and let them have it?
That's called parenting. It is what parents are supposed to do: put the child's best interests first, even if the child wants something else. My cousin would love to eat chips and sausages for dinner every night and watch cartoons all day. Of course she would, she's 4. Does her mom let her? No. She gets that dinner once a week and she spends her days playing and doing physical activities. Has this irreparably damaged my cousin? Of course not. Is my cousin bitter and resentful that she missed her cartoons? Hell no, she will spend dinner time telling you what she did all day. Because she did things all day. And she really enjoys the odd time she is allowed to spend the afternoon watching TV.
Denying a child a donut may upset them, but they will get over it. It's the parents job to decide when these things are appropriate, not the childs. Also OP's description of the child (who is counting calories) leads me to believe that they are more a teenager. That was basically humiliated and bullied into eating a donut s/he didn't want because their mother had a fit.
Let me know if I read your comment incorrectly, but if I am right: WTF? Are you listening to yourself??