T.S.
Although I can see what pp is talking about, I think the analogy is flawed. You should be charged SOMETHING, but not for all three things. In their defense, billing probably just looks at the appointment notes and makes the invoice. The doctor likely doesn't decide/think about WHAT to charge for, and billing doesn't ask. The appointment notes say, IUD insertion and removal, so that's what they bill for.
To me, it would make sense to charge for the visit and insertion, but since you don't actually HAVE the IUD, it doesn't make sense to charge for that. If you don't pick up your prescription, you don't pay for it. Also, since removal wasn't a separate appointment but rather him basically deciding to UNinsert it, you shouldn't be charged for that additionally.
This situation isn't like not paying the allergist because they don't find allergies... it's like the pharmacist spilling bottle on the floor but expecting you to pay for the medication because he tried to fill the Rx.
Call the office and ask. Talk to the doctor directly and see if he can help you with this. Don't assume they're out to get you, come at it from a win-win problem solving mind and you'll get it taken care of.