Is it constitutional? Perhaps the courts would say yes. But that doesn't make the National Day of Prayer a good idea. The Baptist Joint Committee's Brent Walker explains why it's "unnecessary" and "misguided":
"There is nothing wrong with the American people getting together to pray on a designated day, even public officials," Walker said. "In fact every day should be a day of national prayer. . . . The problem with the National Day of Prayer is that it is an official act of the government urging citizens to engage in a religious exercise," Walker said.
…
"A day of prayer is more appropriately called for by pastors, rabbis and imams among us – not civil magistrates, Congress, or even an American president."
You should read the whole statement.
And I will add my own two cents. It's not just a problematic mixing of the government into religion, it's also an unnecessary and harmful politicization of religious expression. Already, groups are criticizing the President for not holding a ceremony in the White House, as if he is somehow slighting religion. It's time we stopped looking to elected officials for an affirmation of our religion, and time we stopped evaluating them for their zeal in doing so.
[UPDATE: As if to provide a perfect example of my complaint, Shirley Dobson – the chair of the National Day of Prayer Task Force – now says of the President not holding a Day of Prayer ceremony in the White House, “At this time in our country’s history, we would hope our President would recognize more fully the importance of prayer.” Mrs. Dobson should not use the pulpit of her congressionally created task force to suggest the President suffers from a lack of religion. If we absolutely must have a National Day of Prayer, would it kill leaders of the Religious Right to leave the politics out of it? Just one more reason it's a bad idea.]