Is it legal for a car insurance company to require medical records?

They can't MAKE you provide them with the information. But they also don't have to provide you with an insurance policy, if you refuse to provide them with the information. The only time they normally do this, is IF you have a medical suspension on your driver's license. That means, a doctor felt strongly enough that you were incapable of driving safely, that they notified the DMV, who put a note on your license. But maybe what happened was that your auto insurance policy cancelled for nonpayment of premium, and you're using the excuse of being in the hospital, of why you didn't pay the bill - in which case, yes, they can "verify" that you're being truthful, as a condition of taking you back. Look, almost anyone can ASK for anything. That doesn't mean you're required to give them what they're asking for. Just don't expect them to give you what YOU are asking for, if you don't comply. It's entirely up to YOU, who you have business transactions with. And it's up to them, too.

They don't want your medical records - just formal confirmation from the hospital that you were there during the period you were not insured. You surely don't expect them to simply take your word for it

Yes, they can ask for private information. You only have to give it to them if you want the insurance. It is a purely voluntary.

So you allowed your insurance to lapse and now want to reinstate it? The insurer probably wants verification that you are physically competent to drive.

They are private. You don't have to show them a thing. But do expect higher premiums if you cannot prove that you were in the hospital

Why would a doctor's note explaining your lapse in coverage by a medical issue be a problem or illegal? You are the one who told then that is what happened.