“It is impossible to look into the Bible with the most ordinary attention without feeling that we have got into a moral atmosphere quite different from that which we breathe in the world, and in the world's literature.”

Washington is known to have made some official statements of public piety, but this is not one of them. The assertion is very widely reported to have been said in Washington's Farewell Address (17 September 1796), but this is not actually the case, as any search of the documents would reveal. It has also been presented as http://www.doctorsenator.com/ReligionandTyranny.html having been part of his Proclamation on January 1, 1795 of February 19th, 1795 as a day of national Thanksgiving. The oldest form of this saying appears as part of an argument for the existence of God attributed to Washington in an undocumented biography written for children. In A Life of Washington (1836) by James K. Paulding, Washington is quoted as having stated:
It is impossible to govern the universe without the aid of a Supreme Being.
(For the context see Paulding's anecdote given below in the section of quotations about Washington.) This is unattributed, and no source other than Paulding is known. In 1864 the words "the aid of a Supreme Being" were replaced by the word "God" in Benjamin Franklin Morris, Christian Life and Character of the Civil Institutions of the United States https://books.google.com/books?id=H92keUU_Xy8C&pg=PA510#v=onepage&q&f=false (1864), p. 510:
:*It is impossible ... to govern the universe without God...
Three years later, in 1867, Henry Wilson (Testimonies of American Statesmen and Jurists to the Truths of Christianity, American Tract Society) replaced "universe" with "world":
:*It is impossible to govern the world without God.
In 1893 Howard H. Russell ( A Lawyer's Examination of the Bible http://books.google.com/books?id=-z0OAAAAYAAJ&pg=PA40#v=onepage&q&f=false, 1893) added the word "rightly" and the phrase "and the Bible" to create the most commonly cited form:
:* It is impossible to rightly govern the world without God and the Bible.
This form, which is also found in Upper Room Bulletin, Vol. 7, No. 3 (23 October 1920) https://books.google.com/books?id=bb_hAAAAMAAJ&pg=PA35&dq=%22It+is+impossible+to+rightly+govern+the+world+without+God+and+the+Bible%22&hl=en&sa=X&ei=whWlVJ61GJDugwTMk4CgDQ&ved=0CDcQ6AEwBA#v=onepage&q=%22It%20is%20impossible%20to%20rightly%20govern%20the%20world%20without%20God%20and%20the%20Bible%22&f=false, rests on no other authority than Russell, who was born long after Washington had died. It is clearly spurious. The saying is often found attached to genuine material such as Washington's 1795 Thanksgiving proclamation http://www.pilgrimhall.org/ThanxProc1789.htm:
:*It is in an especial manner our duty as a people, with devout reverence and affectionate gratitude, to acknowledge our many and great obligations to Almighty God, and to implore Him to continue and confirm the blessings we experienced. It is impossible to rightly govern the world without God and the Bible. It is impossible to account for the creation of the universe, without the agency of a Supreme Being. It is impossible to govern the universe without the aid of a Supreme Being. It is impossible to reason without arriving at a Supreme Being. Religion is as necessary to reason, as reason is to religion. The one cannot exist without the other. A reasoning being would lose his reason, in attempting to account for the great phenomena of nature, had he not a Supreme Being to refer to.
The first sentence is an almost accurate rendition of one from the official proclamation, being a portion of this segment:
: In such a state of things it is in an especial manner our duty as a people, with devout reverence and affectionate gratitude, to acknowledge our many and great obligations to Almighty God and to implore Him to continue and confirm the blessings we experience. Deeply penetrated with this sentiment, I, George Washington, President of the United States, do recommend to all religious societies and denominations, and to all persons whomsoever, within the United States to set apart and observe Thursday, the 19th day of February next as a day of public thanksgiving and prayer, and on that day to meet together and render their sincere and hearty thanks to the Great Ruler of Nations for the manifold and signal mercies which distinguish our lot as a nation...
It is to be noted that there is genuine piety expressed in this statement, but it is not of any sectarian kind, Christian or otherwise. The last portion of the bogus statement which uses it is a truncation of a statement attributed to him in an undocumented biography written for children. In A Life of Washington (1836) by James K. Paulding, Washington is quoted as having stated:
: It is impossible to reason without arriving at a Supreme Being. Religion is as necessary to reason as reason is to religion. The one cannot exist without the other. A reasoning being would lose his reason in attempting to account for the great phenomena of nature, had he not a Supreme Being to refer to; and well has it been said, that if there had been no God, mankind would have been obliged to imagine one.
In the spurious version of the Thanksgiving proclamation which uses a portion of this, Washington's allusions to Voltaire's famous statement that "If God did not exist, it would be necessary to invent him" has been omitted. In the cases of these "quotations" it seems that if statements suitable to their sectarian interests do not exist, some people feel it necessary to invent them.
Misattributed, Spurious attributions