They seem to have gotten it right, under constitutional review. You may not like the law, but they didn’t pass the law, Congress did.
And the constitution clearly gives congress the right to regulate federal elections, which includes deciding HOW people get registered to vote, if congress so chooses, which they did.
We need to petition congress to add a verification for citizenship — good luck getting it through now though.
After reading through several analyses of this decision, it is not nearly as bad as most are portraying it. As several have pointed out, this affects only registration, not requiring ID for voting. And this only impacts registration to vote for federal offices. States are still free to require proof of citizenship to register to vote for state offices, though that would require maintaining a separate registration form and a way to identify those who were registered to vote in Federal elections only, and those who were eligible to vote in state and local elections.
Scalia also laid out a road map for Arizona on how to correct the problem. Basically, Arizona needs to ask the board that controls the makeup of the registration form to include language indicating that Arizona residents need to include proof of citizenship with the registration form. If they refuse, Arizona can sue the board on several grounds (laid out neatly by Scalia), including on the grounds that the board has already granted similar requests to other states.