I have never, ever been asked for proof of anything - not a sick day, not 2 hours off, nothing. Neither has my husband. If you exceed your allotted time, then there's probably a policy about docking you. If you haven't been there long enough to accrue time, then some companies will not pay you and others will log it against personal/sick days you'll be earning over time - and if you left before that, they could take it out of your last check. That only makes sense.
I disagree about HIPAA laws not being violated if you just provide a note that you were in a doctor's office. Providing a letter from a GP/internist is quite different from providing one from a psychiatrist or oncologist or infertility specialist, right? It's not from the patient's point of view, but most people wouldn't want their reason for absence to be obvious by the practitioner's specialty. What if someone's appointment is with a divorce lawyer or a criminal attorney? It's no one's business and I think you should protest that.
The employer has a right to know if you were injured on the job and will be applying for worker's comp. They have a right to know if you aren't coming back or if you will require special accommodations in order to do your job.
Otherwise, your time is your time. Now, if you were to use your sick days on every Friday of the summer and come in on Monday with a great tan, you'd be making it clear that your concern is more with beach weekends than company productivity, and that's a bad idea. If you're magically sick every year on the Monday after the Super Bowl, that's a bad message to send. And I wouldn't use work time to get a haircut or eyebrow waxing or anything else that could easily be scheduled outside of work hours. But attending a kid's music recital is a legitimate use of personal time. You should, obviously, give as much notice as you can in non-emergency situations.
I agree to consult the employee manual or whatever papers you got when you were hired. Some small companies don't have that, and it's just the boss making all decisions, then it's going to be uncomfortable. You should call your state's Division of Employment to get good advice. There's probably a lot of info on line.