The verb nail is used to mean to expose or reveal the falsehood of an allegation, assertion, etc., especially to prevent further dissemination. This use is first recorded in An Oration delivered at the Celebration in Philadelphia of the 106^th Anniversary of the Birthday of Thomas Paine, by John Alberger, published in The North American Review of July 1843:

In the year 1802, Paine returned from France, at the instance of Mr. Jefferson, in a national vessel, and paid his passage by writing for the Administration press abusive essays, which he dignified with the title of letters to the people of the United States. In a letter of the 19^th of November, 1802, [he asserted], by direct implication, that a conspiracy existed between General Washington and Mr. Adams, the first President and Vice-President, to make the Executive office hereditary, descending to Mr. Lund Washington, the President’s nephew, as next of kin, with a sort of contingent remainder to the heirs male of the Vice-President. Such was the incense which a vile party-press offered to its leader, and which his nostrils seemed gratefully to snuff up. We do not stop to nail down this malignant libel.

This use of the verb nail is probably an allusion to a former practice among shopkeepers of nailing counterfeit coins to the counter as a warning or to remind themselves to be watchful (and these coins could be easily compared with others of similar appearance when offered). For example, The Constitution; or, Cork Advertiser (Ireland) of 22nd January 1835 had the following:

Two coins intended for British currency, which were tendered to a Respectable House in Westminster, in payment of a debt due in Cork, but were returned as base, false, and counterfeit, the brass being quite visible—they will do to nail to a counter.

Figuratively therefore, to nail an allegation, assertion, etc., to the counter meant to expose as false or spurious. The Birmingham Daily Post of 20^th October 1888 published the following letter from the British Liberal statesman Joseph Chamberlain (1836-1914):

To the Editor of the Daily Post.
Sir,—In the course of a somewhat incoherent speech to a meeting in St. George’s Ward last night, the chairman, Councillor Whateley, is reported to have used the following words:—“Those gentlemen, some time ago, were so disloyal to her Majesty that on one occasion, when a banquet was given by the Mayor of Birmingham, Mr. Joseph Chamberlain, Mr. Jesse Collings, and Mr. Arthur Chamberlain, if not another member of the family, would not rise with the rest of the company to drink the Queen’s health.”
Allow me at once to nail this lie to the counter.  There is absolutely not the slightest foundation for it so far as I am concerned. I have never refused to drink the Queen’s health, or to stand when it was proposed, and should regard such a refusal as an act of stupid discourtesy.

Joseph Chamberlain’s expression became popular. The Salisbury Times (Wiltshire) of 1^st November 1890 reported that, during a meeting of the local Conservative Club, an orator said:

One of the things the Conservative party had to contend against, one of the greatest hindrances they met with, was the spread of rumours against a candidate or against a member (hear, hear). Let them sift those rumours, find out what was being said behind their backs, and, to use Mr. Chamberlain’s expression,  let them “nail the lie to the counter.”

A variant used mast instead of counter. For example, the following is from The Yorkshire Post of 8^th July 1895:

Addressing a large and enthusiastic open-air meeting at Middlesbrough on Saturday, Colonel S. A. Sadler, the Unionist candidate for the borough, said he had heard some foolish remarks to the effect that his company were paying their workmen an exceedingly poor and beggarly rate of wages. He wished to at once nail that lie to the mast.