Unfortunately for Gawker, they're very likely in the wrong.
Fair use under American statutory and caselaw covers many uses, and if Palin's book were already a published work Gawker's republication of parts of it would be more defensible as an act of fair use. The purpose of their republication of the work was clearly informational with editorial comment, as Gawker would likely be recognized in any lawsuit as a journalistic enterprise and the piece was one of critical opinion, albeit one with a fair amount of brevity. However, the amount republished - twenty-six pages - is certainly towards the large end for a fair use argument, given that it is nearly ten percent of the original work's 304 pages. Further, Gawker's critical comment is restricted to brief and meanspirited summaries of Palin's writing, so the purpose of its use is questionable.
All of that, however, is possibly irrelevant given that Palin's book is not yet technically a published work, since a book is not considered to be published until its release date. Under American common-law rules of fair use, publishing excerpts of a work before its publication date actively harms the case for fair use, as you are considered to be obliviating the creator's rights in publication of the work. The best example of this in American caselaw is Harper & Row v. Nation Enterprises, 471 U.S. 539 (1985), which is largely analagous to the Palin/Gawker incident given that both instances involve an unauthorized excerpt of not-yet-published writing by a public political figure - in Harper it was a memoir by Gerald Ford, excerpted without permission by The Nation magazine. In Harper, the publisher won specifically because the Supreme Court determined that a creator's right of first publication was an important right and any fair use defence would have to be extremely compelling to override it.
(It's worth noting that the online fair use summary Gawker linked to when they sarcastically mocked Palin's legal knowledge actually refers to this case specifically in discussing when something is and is not fair use.)
Interestingly, were this incident to occur in Canada, fair dealing (the Commonwealth equivalent of fair use) a finding of fair dealing might be more likely, given that under Canadian caselaw - specifically CCH Canadian Ltd. v. Law Society of Upper Canada, which sets out most of the modern common-law rules for fair dealing in a Canadian context - excerpts from an unpublished work are more likely to be considered fair than excerpts from a published work, given that an unauthorized reproduction could potentially lead to a wider public dissemination of the work, which is ostensibly one of the goals of creation in the first place. This is one of the major differences between Canadian and American law on fair use/fair dealing.