Is there a database that matches each US phone area code to the major city and state that it belongs to?
The North American Numbering Plan Administration is responsible for the number resources. On their website they have both a Flash and an HTML method of looking at the area codes by state. You would need to manually analyze the data, but this is the best source to answer your question.
There are also excel spreadsheets of the area codes, but it does not have the city information you are looking for.
Below is the area codes for Florida, you can see the major city and state information per your request.