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.
Best Answer
Google needs to integrate Siri into their Google Voice app. Only if Google makes the integration you can use Siri with Google Voice. "Call Alice with Google Voice" is a good example.
Audio or video calling integration with Siri is possible since iOS 10 (September 2016). And for developers to build the integrations since June 2016.
So my best advice is, start asking Google to build this integration.