It's already functionally true - you need to be a citizen to vote, and in order to register to vote, you have to present multiple forms of ID. The problem is that we don't have a unified national ID system, so forms of ID that are accepted or required can vary state to state, and they're not always something that every person has easy access to, so it's become somewhat of a thorny issue. It's compounded by the fact that voter ID laws have been used to suppress voting in certain communities (most notably the African-American community following the Civil War, because slaves weren't given birth certificates or recognized as citizens until after the war (and we underserved for many decades afterwards), and thus often had great difficulty proving their identities. It's much less of an issue now , but there's baggage attached to it still.
If we had a unified national ID that was issued for free to every citizen at birth, I think very few people would have an issue with requiring that that ID be presented in order to vote, but for whatever reason, those who argue most strongly for requiring voters to show ID tend to be deeply opposed to a national ID system.