Hello, and I hope I can help you today.
I agree with you that the decision to end your wife's induction is not the safest idea. If she truly has severe
preeclampsia (which is what the headache is from) it does not go away even if her pressure stabilizes on
bed rest.
However, I assume they did not break your wife's water already as part of the induction, because of her water is broken she absolutely should not go home from the hospital.
If they never broke your wife's water, they never really committed to inducing her labor. It can take even two days to induce someone at 37 weeks if they are not dilated at all.
So, and I preface this by saying I don't know your wife's full medical condition, but based on what you are telling me I think that you do not need a c-section right now, but I think they have to at least try to break her water, and if she is not dilated enough to do that after 2 days induction, a c-section is a reasonable decision.
I hope that I adequately answered your question and that my advice was helpful.
Best wishes for the delivery and your wife and baby,
Dr. Brown