As others have said, you do not need internet for GPS.
GPS is basically a satellite based positioning system that is designed to calculate geographic coordinates based on timing information received from multiple satellites in the GPS constellation. GPS has a relatively slow time to first fix (TTFF), and from a cold start (meaning without a last known position), it can take up to 15 minutes to download the data it needs from the satellites to calculate a position. A-GPS used by cellular networks shortens this time by using the cellular network to deliver the satellite data to the phone.
But regardless of whether it is an A-GPS or GPS location, all that is derived is Geographic Coordinates (latitude/longitude). It is impossible to obtain more from GPS only.
To be able to return anything other than coordinates (such as an address), you need some mechanism to do Reverse Geocoding. Typically this is done by querying a server or a web service (like using Google Maps or Bing Maps, but there are others). Some of the services will allow you to cache data locally, but it would still require an internet connection for periods of time to download the map information in the surrounding area.
While it requires a significant amount of effort, you can write your own tool to do the reverse geocoding, but you still need to be able to house the data somewhere as the amount of data required to do this is far more you can store on a phone, which means you still need an internet connection to do it. If you think of tools like Garmin GPS Navigation units, they do store the data locally, so it is possible, but you will need to optimize it for maximum storage and would probably need more than is generally available in a phone.
Bottom line:
The short answer to your question is, no you do not need an active internet connection to get coordinates, but unless you are building a specialized device or have unlimited storage, you will need an internet connection to turn those coordinates into anything else.