I think the spec you're looking for is the FileSystem API (https://developer.mozilla.org/en-US/docs/Web/API/File_and_Di...), not the File API.
Unfortunately, this is a non-standard API implemented only by Chrome. You have to use other APIs like IndexedDB and WebSQL (also deprecated and non-standard) to get a working solution in all browsers.
This deficiency is really holding back the web.