The answer to the question you asked in another thread ([POLL] Shorten the 5 minute disconnection timer - #44 by Atorrante) is no, the U.S. does not have anything comparable to the French Academy. For spelling, people refer to any large dictionary, such as Webster’s 2nd edition, Merriam Webster, Random House Unabridged, etc.
Instead of an authoritative government body to dictate style in American English, we use one of five standard manuals to guide us, depending on the context: Chicago (University of Chicago; for scholarly work in the humanities), APA (American Psychological Association; for scientific research), GPO (Government Printing Office; for government work), AP (Associated Press; for newspapers, magazines, etc.), and MLA (Modern Languages Association; for papers in college and high school). In addition, many specialized communities have their own style manual; for example, the military Services each has its own particular style guide. The same thing occurs among specialized publications, where such guides are called “house style.”
Style consists of five levels of increasing difficulty: punctuation, grammar, usage, syntax, and narrative. The style manuals, grammar books, and Words into Type cover punctuation and grammar. For the higher levels of style, my favorite books are Fowler’s Modern English Usage (first edition only), Follett’s Modern American Usage, Herbert Spencer’s Philosophy of Style (really just a long essay), and Gower’s The Complete Plain Words. I don’t know of a good book on syntax; the one I have is so technical and jargon ridden that it is virtually unreadable (lol). Narrative is covered by many books about writing fiction or nonfiction, but a proper sense of narrative is gained best through extensive reading.