The earliest dictionary entry that I could find for the word spells it neither public nor publick, but publique. From Henry Cockeram, The English Dictionarie: Or, an Interpreter of Hard English Words, second edition (1626):
Publique, Common, open.
This dictionary persisted in its preference for publique at least through the seventh edition (1642), but by the ninth edition (1650) it had switched preference to publick:
Publick, Common, open.
Most subsequent seventeenth-century dictionaries didn't consider the word sufficiently hard to merit its own entry, but they used the spelling publick in definitions of other words. And Elisha Coles, An English Dictionary Explaining the Difficult Terms That Are Used... (1676) offered a definition for "publick faith":
Publick Faith, on which the Parliament rais'd money to carry on the war.
Coles's editions of 1692 and 1717 retained this definition and spelling of "publick faith."
The early eighteenth century saw the emergence of general English dictionaries—lexicons that, unlike their predecessors were not limited to a focus on difficult words. For most of the 1700s, publick was the dominant spelling used in these dictionaries. Thus, for example, John Kersey, Dictionarium Anglo-Britannicum: Or, A General English Dictionary (1708) offered these entries for the word:
Publick, common, belonging to the Poeple ; manifest, known by every Body
The Publick, the generality of People.
Editions of Kersey's various dictionaries retain these entries (with improved spelling of "people") at least as late as A New English Dictionary: or, A Compleat Collection of the Most Proper and Significant Words, and Terms of Art Commonly Used in the Language, seventh edition (1757); all use the spelling publick.
Nathan Bailey, Universal Etymological Dictionary, second edition (1724) repeats Kersey's entries and definitions for publick verbatim, adding only etymological notes linking the word to French Public and Latin Publicus; the same entries were in place five decades later in Bailey's twenty-third edition (1773).
Another major dictionary of the middle 1700s, Thomas Dyche & William Pardon, A New General English Dictionary (1735) has this entry for publick:
PUBLICK (A.) Common, open, free for every Body's Use, &c.
This entry remained unchanged in subsequent editions of Dyche & Pardon through the eighteenth edition (1781).
That brings us to Samuel Johnson, A Dictionary of the English Language (1755), which devotes almost a full column to various definitions of and citations using publick as an adjective, publick as a noun, publickly (an adverb), publickness (a noun) and publickspirited (as adjective). Although Johnson notes quotations from Granville, Swift, Milton, Clarendon, Atterbury, Addison, and Pope that use the spelling public, he does not suggest it as an alternative spelling in good contemporaneous standing.
The many revisions of Johnson's Dictionary hew to the spelling publick for most of the next century. As late as 1839, companies were publishing versions of the Todd revision of Johnson's Dictionary with publick in place as the standard spelling for that word:
Publick a. common, not private, manifest.
Publick, s. the body of a nation; the people.
Support for the spelling publick in late eighteenth-century dictionaries was not unanimous, however. A notable exception to the prevailing preference appears in John Marchant, A New Complete English Dictionary, second edition (1760):
PUBLIC (A.) manifest, common, known by every body, open, notorious, generally known.
PUBLIC. (S.) the general body of mankind, or of a state or nation ; open view ; general notice.
The same entries—and the same spelling of public—appear in Marchant's fourth edition (1764).
Another dissenter from the form that Johnson supported was John Ash, The New and Complete Dictionary of the English Language (1775):
PUBLIC (adj. from the Lat. publicus) Open, notorious, generally known ; open for general entertainment ; general, done by many ; belonging to the state, opposed to private.
Public (s. from the adj.) The general body of the people; open view, general notice.
In his introduction to the first volume of his dictionary, Ash discusses this orthographical change:
The final k after c, in words derived from the learned languages, though carefully retained by Johnson and other writers, has been omitted, in conformity to modern custom and the originals. For it seems to me to be rather incongruous to write musick from musica, especially as the k has been exploded by general consent from the derivations musician and musical.
In the United States, Noah Webster seems to have been in full agreement with Ash on this point. Writing in the introduction to A Compendious Dictionary of the English Language (1806), Webster says this:
I have made no material alterations in the orthography of words, except to correct most palpable errors. In a few instances, I have preferred the orthography of Newton, Prideaux, Hook, Dryden, Whiston, &c. to that of Johnson, as being more analogical and purely English, as in scepter, sepulcher. In omitting u in honor and a few words of that class, I have pursued a common practice in this country, authorized by the principle of uniformity and by etymology, as well as by Ash's dictionary. In omitting k after c, I have unequivocal propriety , and the present usage for my authorities.
Indeed, as Webster observes, Ash's dictionary spells honor without a u, asserting in a parenthetical note that honor is "a modern but correct spelling, from the Lat." This suggests that the move toward public from publick was, in part, an element in a broader prescriptive orthographical movement drawing on etymological considerations—with support from lexicographers on both sides of the Atlantic.
By the middle 1800s, the tide had clearly turned against publick. One sign of this is the switch in Todd's revision of Johnson's Dictionary from publick in the 1839 edition (noted above) to public in the 1843 edition. When Johnson's own posterity abandoned his spelling, publick was clearly doomed.
Both P. Austin Nuttall's 1855 revision of Johnson's Dictionary and James Murray's 1874 revision of it likewise use public to the exclusion of publick.
The shift away from publick and toward public may well have been primarily organic‚that is, a matter of popular choice, despite the preference for publick expressed by most English dictionaries from the late 1600s until at least 1760. Nevertheless, to judge from the quotations that Johnson cites in his 1755 dictionary, many eminent authors before Johnson's time used the spelling public at least occasionally.
Writing in 1775, John Ash asserts that omission of the final k is "in conformity to modern custom and the originals"—an explanation that mashes together an argument based on contemporaneous usage and an argument based on historical propriety (presumably on the basis that the French and Latin sources of words such as public and music did not include a k). In the United States, from 1806 forward, Noah Webster subscribes to the etymological argument for public, too. It's difficult to gauge how much (if at all) individual writers were inclined to adjust their spelling preferences on the basis of etymological considerations, but it is certainly interesting that as a group they gradually broke with Johnson despite his authoritative standing in the late eighteenth century.
Despite the movement in popular usage and the revisionist views of of Marchant, Ash, and Webster, various dictionaries based on Johnson's—including Thomas Sheridan, A Complete Dictionary of the English Language, second edition (1789)—preserve Johnson's preference for publick throughout the remainder of the eighteenth century and well into the nineteenth century. The last instance I could find of publick as the preferred dictionary spelling for the word is in a U.S. edition of Todd's revision of Johnson's Dictionary published in 1839.