Wrong. "It's" is incorrect here.
"Its" is a possessive pronoun, along with yours, his, hers, theirs, etc... None of which use an apostrophe. The confusion arises from apostrophes used after a noun, such as "RedAndWhiteBaron's grammar is beyond reproach" - because "it" is a noun but "its" is a pronoun. "The dog's bone" is also correct, as "dog" is a noun. However, "It's bone" is not correct in this context. It would be correct, though, if used to mean "It is [made of] bone".
In general apostrophes are only used with nouns to indicate the possessive, not pronouns, and the only correct use of "it's" is as a contraction for "it is", "it has", or "it was", and even "it was" is rather an edge case.
I do love English though. Sooooo much fun to argue over it. Sorry, my father learned English as a second language and fell in love with it. I had endless suppertime debates about the language with him in his effort to better speak and read/write it. So unfortunately, I now love arguing over English, mostly because I'm alway's right.
Yeah sorry about that. I can't resist.