The History of Keeping Women from Exercising
A friend of mine was told she couldn't train judo because it would somehow harm her reproductive organs. (At no time was she asked whether this mattered to her.) We know that even as recently as the 1970s, women were barred from marathons on similar grounds:
I'm sure many physicians trained in that era are still spouting the same horseshit. Exercise-induced infertility is one of the oldest anti-woman canards in the history of misogyny. Alarm bells should go off when we hear it.
Does it Hold Water?
How many scaremongering news stories do you hear about male infertility caused by judo, or sit-ups, or football? Practically none, despite the fact that the testes are much more exposed to external injury and are damaged by overheating. Likewise, how many stories of infertility (or uterine prolapse) do you hear among the women who lift weights, run marathons, do step aerobics, play judo, or CrossFit? How many incidents are reported nationwide each year? Millions of women train hard. We would be seeing an epidemic if these activities carried significant risks.
Another answer claims that sit-ups cause uterine prolapse, which is a serious condition. The idea is that sit-ups increase intra-abdominal pressure, pushing the uterus down into the vagina. What is left unmentioned is the degree to which this is a risk factor compared to the other causes: "aging, time-related weakening of the ligaments holding the uterus up; having had vaginal deliveries; and a drop-off in estrogen production".
What else causes intra-abdominal pressure? A vast array of common activities, not just hard exercise. Something as everyday as taking a poop is sufficient. The article warns against sit-ups for this woman--not all women--because she recently experienced a uterine prolapse. This does not contraindicate sit-ups for women. That's just plain put-the-woman-back-in-the-kitchen scaremongering BS.
Similarly, the idea that sit-ups cause infertility strikes me as silly. They're not a particularly rough exercise. Are women's ovaries really that weak? I think not.
In contrast, we do see that extensive exercise can, for some women, in some circumstances, cause irregularities in menstrual cycles. (It should be noted that not exercising will of course have negative effects as well.) A brief PubMed search resulted in nothing suggesting that sit-ups are particularly harmful towards women's health.
Sit-ups are dumb anyway
The reason to avoid sit-ups is not for any of these anti-woman tropes, nor nonexistent dangers. It's because the exercise itself just isn't too great. Sit-ups are not that efficient at training the abs, and they can put stress on the lower back.
More importantly, spot reduction is a myth, so it won't help reduce belly fat any more than squats or bicycling would.
Your friend shouldn't do sit-ups, but she shouldn't do them because she's too busy lifting heavy weights, running, or boxing, not because they constitute some danger to her frail womanly nature.