What happens on the national day of prayer is that private organizations organize prayer breakfasts or other events for the purpose of praying for the United States. Government proclamations of this nature tend to recognize community events that have been organized without them rather than declaring that everyone has to observe them. If there's a national ice cream day, it doesn't mean you have to acknowledge it in any way or eat ice cream. Nor is it a tacit government endorsement of the activity.
The National Day of Prayer Task Force is run by Focus on the Family, so most events that are organized fit that bent and are done so privately. I don't think many governors are setting up events in their state themselves, though they also tend to issue proclamations acknowledging the day. I'm pretty sure these events would happen with or without a presidential proclamation. They might change the name to 'Day of Prayer for our Nation' though, or something
Here's snippets of what the proclamations have said (under Reagan and Clinton):
1988: "On our National Day of Prayer, then, we join together as people of many faiths to petition God to show us His mercy and His love, to heal our weariness and uphold our hope, that we might live ever mindful of His justice and thankful for His blessing."
1993: "Through prayer our people take a moment away from the concerns of everyday life to understand the greater power that gives us guidance. We come together in an act common to all religions."
1994: "I encourage the citizens of this great Nation to gather, each in his or her own manner, to recognize our blessings, acknowledge our wrongs, to remember the needy, to seek guidance for our challenging future, and to give thanks for the abundance we have enjoyed throughout our history."
1995: "I call upon every citizen of this great Nation to gather together on that day to pray, each in his or her own manner, for God's continued guidance and blessing."
1996: "And though our citizens come from every nation on Earth and observe an extraordinary variety of religious faith [sic] and traditions, prayer remains at the heart of the American spirit."
1997: "...let us uphold the tradition of observing a day in which every American, in his or her own way, may come before God seeking increased peace, guidance, and wisdom for the challenges ahead."
1998: "In every city, town, and rural community across our country, people of every religious denomination gather to worship according to their faith. In churches, synagogues, temples, and mosques, Americans come together to pray."
And from Lincoln and Obama:
By the President of the United States of America.
A Proclamation.
Whereas, the Senate of the United States, devoutly recognizing the Supreme Authority and just Government of Almighty God, in all the affairs of men and of nations, has, by a resolution, requested the President to designate and set apart a day for National prayer and humiliation.
And whereas it is the duty of nations as well as of men, to own their dependence upon the overruling power of God, to confess their sins and transgressions, in humble sorrow, yet with assured hope that genuine repentance will lead to mercy and pardon; and to recognize the sublime truth, announced in the Holy Scriptures and proven by all history, that those nations only are blessed whose God is the Lord.
And, insomuch as we know that, by His divine law, nations like individuals are subjected to punishments and chastisements in this world, may we not justly fear that the awful calamity of civil war, which now desolates the land, may be but a punishment, inflicted upon us, for our presumptuous sins, to the needful end of our national reformation as a whole People? We have been the recipients of the choicest bounties of Heaven. We have been preserved, these many years, in peace and prosperity. We have grown in numbers, wealth and power, as no other nation has ever grown. But we have forgotten God. We have forgotten the gracious hand which preserved us in peace, and multiplied and enriched and strengthened us; and we have vainly imagined, in the deceitfulness of our hearts, that all these blessings were produced by some superior wisdom and virtue of our own. Intoxicated with unbroken success, we have become too self-sufficient to feel the necessity of redeeming and preserving grace, too proud to pray to the God that made us!
It behooves us then, to humble ourselves before the offended Power, to confess our national sins, and to pray for clemency and forgiveness.
Now, therefore, in compliance with the request, and fully concurring in the views of the Senate, I do, by this my proclamation, designate and set apart Thursday, the 30th. day of April, 1863, as a day of national humiliation, fasting and prayer. And I do hereby request all the People to abstain, on that day, from their ordinary secular pursuits, and to unite, at their several places of public worship and their respective homes, in keeping the day holy to the Lord, and devoted to the humble discharge of the religious duties proper to that solemn occasion.
All this being done, in sincerity and truth, let us then rest humbly in the hope authorized by the Divine teachings, that the united cry of the Nation will be heard on high, and answered with blessings, no less than the pardon of our national sins, and the restoration of our now divided and suffering Country, to its former happy condition of unity and peace.
In witness whereof, I have hereunto set my hand and caused the seal of the United States to be affixed.
Done at the City of Washington, this thirtieth day of March, in the year of our Lord one thousand eight hundred and sixty-three, and of the Independence of the United States the eighty seventh.
By the President: Abraham Lincoln
William H. Seward, Secretary of State
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THE WHITE HOUSE
Office of the Press Secretary
For Immediate Release May 7, 2009
NATIONAL DAY OF PRAYER, 2009
BY THE PRESIDENT OF THE UNITED STATES OF AMERICA
A PROCLAMATION
Throughout our Nation's history, Americans have come together in moments of great challenge and uncertainty to humble themselves in prayer. In 1775, as the Continental Congress began the task of forging a new Nation, colonists were asked to observe a day of quiet humiliation and prayer. Almost a century later, as the flames of the Civil War burned from north to south, President Lincoln and the Congress once again asked the American people to pray as the fate of their Nation hung in the balance.
It is in that spirit of unity and reflection that we once again designate the first Thursday in May as the National Day of Prayer. Let us remember those who came before us, and let us each give thanks for the courage and compassion shown by so many in this country and around the world.
On this day of unity and prayer, let us also honor the service and sacrifice of the men and women of the United States Armed Forces. We celebrate their commitment to uphold our highest ideals, and we recognize that it is because of them that we continue to live in a Nation where people of all faiths can worship or not worship according to the dictates of their conscience.
Let us also use this day to come together in a moment of peace and goodwill. Our world grows smaller by the day, and our varied beliefs can bring us together to feed the hungry and comfort the afflicted; to make peace where there is strife; and to lift up those who have fallen on hard times. As we observe this day of prayer, we remember the one law that binds all great religions together: the Golden Rule, and its call to love one another; to understand one another; and to treat with dignity and respect those with whom we share a brief moment on this Earth.
The Congress, by Public Law 100-307, as amended, has called on the President to issue each year a proclamation designating the first Thursday in May as a "National Day of Prayer."
NOW, THEREFORE, I, BARACK OBAMA, President of the United States of America, do hereby proclaim May 7, 2009, as a National Day of Prayer. I call upon Americans to pray in thanksgiving for our freedoms and blessings and to ask for God's continued guidance, grace, and protection for this land that we love.
IN WITNESS WHEREOF, I have hereunto set my hand this seventh day of May, in the year of our Lord two thousand nine, and of the Independence of the United States of America the two hundred and thirty-third.
BARACK OBAMA
(The third Sunday in July is Nat'l Ice Cream Day, as proclaimed by Ronald Reagan in 1984, in case you were curious.)