Previously asked for iOS 7. I need to use Google Voice (e.g. for cheaper overseas calls), not just for the Caller ID part of the problem.
iOS 10+ is fine if that solves the problem.
To the close vote: I wish to be able to use Siri and say "Call Alice" and the call go via Google Voice. Or "Call Alice with Google Voice" would be good too. I did ask the local Apple Genius Bar. And since (as of 2017-11-01), Google Voice isn't a supported 3rd-party Siri integration, that doesn't work.