I've seen native iOS applications such as Instagram open up a mobile Safari window when they need you to approve them to use your Facebook or Twitter account. Once you're done, the mobile Safari window closes, and you are taken back to the app automatically.
Is there any way to do this from a homescreen web app? Or failing that, a Phonegap app?
Unfortunately not from a homescreen web app.
However, it should be entirely possible from a Phonegap app, as this appears to be a 'native wrapper' around your web application.
Usually twitter/facebook ask for some kind of 'callback url' to facilitate this.
On iOS you can register a URL scheme for your application, such as "myapp://". Providing said URL scheme as the callback url should cause your application to be opened.
Take a look at the following tutorial.
