Paying for hospital in Nice

Have your bank wire payment to them. They will need the special bank code for the medical facility. I have done this to pay traffic tickets which arrived after returning home.
I also was treated in an ER in Europe and was given the bill before I left, much easier. The costs are so much lower there than in the US.

There is a website some folks here have used for paying European traffic tickets. It has been mentioned at least two or three times. Unfortunately, I don't remember the name of it. Reports indicated that it was much, much less expensive than arranging a wire transfer through a bank.

Somewhere on the bill will be the "IBAN" (=International Bank Account Number), It will start with "FR" (for France).
If you have an account at a bank in most countries of Europe, normal on-line banking allows you to send money to an IBAN at little or no cost. This is the method of payment their system is expecting.

If you live in a country that doesn't use IBAN, you will have to go to your bank and get them to send the money to the IBAN manually. This may be difficult (as the staff may not know about IBAN's), and it may cost you.

Yes wire transfers through your bank will be expensive plus they will ding you with a foreign currency transaction fee. It could cost as much as the bill.

I would email them first and ask if you can pay with a credit card which you can email them the number, expiration date, CCV number as well as the the name and address of the credit card holder. Even offer to pay a 5% fee on top for their trouble. You can also ask if they take Paypal.

Email will work better as you both will have written record of the conversation, and it will be easier for them to translate it. And be sure to thank them for their service. I doubt if you could visit an ER in the States any where near as low as $100 without an insurance card.

shouldn't the starting point be asking the hospital what methods of payment they accept (maybe the bill states this)

reply to the email you received and ask your questions - they are best placed to advise

Be careful about sending credit card info through email, it's pretty public. Especially if you include the CVV code, then someone is off to the races.

I was recently dealing with a restaurant in Boston for some catering and they quite explicitly said DO NOT GIVE US YOUR CC INFO IN AN EMAIL! They said either call it in or send a fax (how quaint).

You can send the information split into separate emails. I'm more suspicious of servers in Boston than I am of hospital administrators in France. I've done it several times. Credit card fraud is so easy to spot, especially if you have an outfit like Capital One watching. First inkling and bang, you get a new credit card. But you can work that out when you've figured out a payment method.

Be it email, fax, or telephone there is no secure way of sending CC numbers; which is why you are protected from fraudulent charges. Lots of fraud comes from very low-tech methods like sifting trough garbage cans, stealing postal mail, or a dishonest employee simply writing down the number on a piece of paper and selling it off to a criminal group.
In your shoes I would email the credit card number. Odds are nothing bad will happen, If something bad does happen your bank will cancel the bad charges, and issue you a new card....life will go on;)

If you're concerned about credit card fraud, you might consider purchasing a VISA gift card with sufficient funds to cover the bill and any international charge. You will pay a small fee (about $4 for a VISA gift card loaded with $100) but you can simply toss the card after using it, with no risk of anyone making fraudulent charges because there will be no credit remaining on the card.

To combat money laundering by drug cartels Visa gift cards can now only be used within the country they were purchased.

The company that will do a wire transfer much cheaper than a US bank is Transferwise. Here's the post, with the satisfied follow up: https://community.ricksteves.com/travel-forum/budget-tips/alternative-to-bank-wire-transfer-transferwise

I agree, first try e-mailing them to ask about methods of payment. Even if the person who gets the e-mail doesn't speak English, they can take it to someone else for assistance.