I feel like Conor is being a little coy with this.
First off, the Declaration of Independence has no binding legal effects on the United States, so there's no reason to Amend it. Second, if the idea is to demonstrate the Christian origins of our nation, that's a famously difficult question, as many of our founding fathers publicly and explicitly denied the divinity of Jesus Christ. The constitution, the actual document outlining federal governance, is explicit in its denial of a state-established religion. Perhaps germane to the discussion is our treaty with Tripoli, written by the Washington administration and signed by John Adams, which states "the government of the United States is not in any sense founded on the Christian religion."
Look, depending on how you define the question, you can answer affirmatively that the United States is a Christian nation, just in terms of the history of having a dominant Christian majority. But in terms of the legal establishment of a state religion, we should privilege what our basic governing document actually says. Now, like I said in the post in question, I am little moved by things like Christmas trees in the town green or similar, though, no, in a perfect world I would probably remove such symbols. I find "in god we trust" on money and that sort of thing pointless religious symbolism where it doesn't belong. (Who is served, exactly, by that little bit of writing? Unless the god we're talking about is money....) Again, though, as a matter of political will, I'm not compelled to make it a big deal, although my instinct is that one day, such symbols will disappear-- not because of a death of religiosity but because of a healthy respect for the division of church and state.
Conor's other concerns seem like weak brew to me. I certainly don't think we should stop having holidays, but we could easily make Christmas a "winter celebration" day. Again, that'd be way down on my list of priorities, and that should wait until we as a country have more comfortably separated religion and government. It isn't, I think, a monumental task to preserve the actual observation of federal holidays while removing the religious trappings.
Do I want to pull the accreditation of parochial schools? Not at all. But they need to pass the same basic process of validating their educational value, and they must not receive any government money, in any way, shape or form. Government has a duty to respect the right of parents to put their children into accredited schools of any tradition; it has absolutely no duty to pay the way of parents who don't like the public schools. As far as religious charities go, I actually said this explicitly in the previous post. Provided that a religious charity fulfills the same requirements of charitable giving a secular charity does, they might receive federal money. As far as special religious use of illegal drugs, that's kind of a moot point for me, as I support the legalization of all drugs on libertarian grounds.
As far as the First Amendment goes, I'm not getting the same reading Conor is. The constitution already protects expression and the right to gather. It seems clear to me that saying that you can't abridge the freedom to practice a particular religion, while preventing the privileging of any particular tradition, leads to exactly the kind of distance between church and state I'm looking for. The government can't come in and say "no Quakers", but neither can it come in and say "Quakers above all". I don't think the constitution establishes any special privileges for religion at all. It merely explicitly forbids the kind of religious oppression that had been common in Europe.
To me, the heartening thing about all of this is that the divide between a secular sphere of government and a religious private sphere is that it seems very natural, and in keeping with the American character. People would continue to go to their houses of worship and pursue their religious inclinations in their homes. They could still meet for fellowship and worship with their peers. But they would do so understanding that it would inappropriate for the government to sanctify or privilege their observation in any way, and would expect no recognition of it in the civic sphere. That permits a robust and functioning religious community while ensuring that religious arguments and religious trappings have no place inside government.
Update: Seems Alex Knapp at Heretical Ideas beat me to most of this.