A name is not a requirement to render medical service, so I don't see how that example is relevant. A practioner is capable of treating patients without knowing their name. Laws may compel them to keep track of that data, but it's not strictly necessary.
And the act of connecting to a server hosted in another jurisdiction (e.g. America) would require sharing your IP. This could be directly (the entire web service hosted in the USA), or indirectly (some of the web service's assets are hosted in the USA).
If you put a CDN in-front of your web service, then that CDN will most likely be sharing your IP with the host server too. Especially if the web service wants to do something non-cacheable that they can't offer from behind the CDN.