Yes, the phrase used is often, "By the power vested in me by XYZ".
Marriage, particularly in the US has been a civil contract for as long as I can find records. You can get a civil only wedding, but not one that is just religious.
The license is from the state.
It sets out privilages and responsibilities involved in the contract.
That is why I find it so hard to understand why anyone opposes same-sex marriages.
Aslan is not a Tame Lion