Given that

a) the secret ballot was introduced into the US (originally as the "Australian ballot") many years after the ratification of the Constitution b) no amendment has prescribed it

I find it hard to quarrel with the judge. I do consider the secret ballot an excellent idea, but I don't see it as constitutionally mandated.

You guys really should have copied preferential voting while you were at it. Or skipped over us entirely and gone with proportional representation.