“I see great things in baseball, It will take our people out-of-doors, fill them with oxygen, give them a larger physical stoicism, tend to relieve us from being a nervous, dyspeptic set, repair those losses and be a blessing to us.”
This has been widely attributed to Whitman, and no one else, but without definite source. It has sometimes been cited as being from the Brooklyn Daily Eagle (sometimes with a date of 23 July 1846), where Whitman had been an editor, but its presence on that date is not apparent in the online historical archives http://www.brooklynpubliclibrary.org/eagle/ of that publication.
Brian Cronin, in "Did 'Bull Durham' misquote Walt Whitman on baseball?" http://www.latimes.com/sports/sportsnow/la-sp-sn-bull-durham-baseball-20120328,0,5200453.story, Los Angeles Times (28 March 2012), suggests that this is (loosely) paraphrased from a remark of September 1888 reported in Horace L. Traubel, With Walt Whitman in Camden, Vol. 2:
I like your interest in sports ball, chiefest of all base-ball particularly: base-ball is our game: the American game: I connect it with our national character. Sports take people out of doors, get them filled with oxygen generate some of the brutal customs (so-called brutal customs) which, after all, tend to habituate people to a necessary physical stoicism. We are some ways a dyspeptic, nervous set: anything which will repair such losses may be regarded as a blessing to the race. We want to go out and howl, swear, run, jump, wrestle, even fight, if only by so doing we may improve the guts of the people: the guts, vile as guts are, divine as guts are!
"Sports for a Dyspeptic Race", Intimate With Walt: Whitmans Conversataions With Horace Traubel, p. 261 https://books.google.com/books?id=_Rp_4VHeQkAC&printsec=frontcover&dq=With+Walt+Whitman+in+Camden&hl=en&sa=X&ei=dqMtVfHQLcODsAWM-ICIDQ&ved=0CEUQ6AEwBQ#v=onepage&q=base-ball&f=false
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Walt Whitman 181
American poet, essayist and journalist 1819–1892Related quotes

ME 13:420
1810s, Letters to John Wayles Eppes (1813)

"The Film Foundation Main page".

“Difficulties give us the opportunity to prove our greatness by overcoming them.”
Message dictated to Sanjeeva Reddy at Guruprasad (6 June 1960), as quoted in The God-Man : The Life, Journeys and Work of Meher Baba with an Interpretation of His Silence and Spiritual Teaching (1964) by Charles Benjamin Purdom, p. 353 <!-- also quoted in The Silent Master : Meher Baba, Avatar of the Age (1987), by Irwin Luck, p. 15 -->
General sources
Context: It is better to deny God, than to defy God.
Sometimes our weakness is considered strength, and we take delight in borrowed greatness.
To profess to be a lover of God and then to be dishonest to God, to the world and to himself, is unparalleled hypocrisy. Difficulties give us the opportunity to prove our greatness by overcoming them.

Letter to his daughter Sarah Mason McCarty after the death of an infant daughter (10 February 1785), published in The Life of George Mason, 1725-1792 Vol. 2 (1892) by Kate Mason Rowland, p. 74

As quoted in NEA Journal : The Journal of the National Education Association Vol. 41 (1952) p. 300
Context: One of the basic causes for all the trouble in the world today is that people talk too much and think too little. They act too impulsively without thinking. I am not advocating in the slightest that we become mutes with our voices stilled because of fear of criticism of what we might say. That is moral cowardice. And moral cowardice that keeps us from speaking our minds is as dangerous to this country as irresponsible talk. The right way is not always the popular and easy way. Standing for right when it is unpopular is a true test of moral character. The importance of individual thinking to the preservation of our democracy and our freedom cannot be overemphasized. The broader sense of the concept of your role in the defense of democracy is that of the citizen doing his most for the preservation of democracy and peace by independent thinking, making that thinking articulate by translating it into action at the ballot boxes, in the forums, and in everyday life, and being constructive and positive in that thinking and articulation. The most precious thing that democracy gives to us is freedom. You and I cannot escape the fact that the ultimate responsibility for freedom is personal. Our freedoms today are not so much in danger because people are consciously trying to take them away from us as they are in danger because we forget to use them. Freedom unexercised may be freedom forfeited. The preservation of freedom is in the hands of the people themselves — not of the government.
Source: "Drug Approved. Is Disease Real?" https://www.nytimes.com/2008/01/14/health/14pain.html?pagewanted=all&_r=0, New York Times (14 January 2008)