"Well, for starters, I came to that conclusion because the Pope didn't say a State must declare Jesus is Lord."
Does the religion "established by Jesus Christ Himself, and which He committed to His Church to protect and to propagate." declare Jesus is Lord?
So if the state must profess the true religion, which declares Jesus is Lord, then the state must declare Jesus is Lord.
Liberals like Neuhaus like to pick and choose their quotes, always with an eye towards forwarding their liberal agenda. So before someone brings up "Pope Leo was speaking to another time" or some other such nonsense (as if truth changes). The often vague Catechism of the Catholic Church, released less than 20 years ago says exactly the same thing:
2105 The duty of offering God genuine worship concerns man both individually and socially. This is "the traditional Catholic teaching on the moral duty of individuals and societies toward the true religion and the one Church of Christ."30 By constantly evangelizing men, the Church works toward enabling them "to infuse the Christian spirit into the mentality and mores, laws and structures of the communities in which [they] live."31 The social duty of Christians is to respect and awaken in each man the love of the true and the good. It requires them to make known the worship of the one true religion which subsists in the Catholic and apostolic Church.32 Christians are called to be the light of the world. Thus, the Church shows forth the kingship of Christ over all creation and in particular over human societies.33
And in fact the "33" in the above quote refers to this footnote:
33 Cf. AA 13; Leo XIII, Immortale Dei 3,17; Pius XI, Quas primas 8,20.
Sorry, no way out for Fr. Neuhaus. He is denying the catechism. While it does not necessarily rise to the level of formal heresy, it is material heresy.