Let me put this here for the sake of clarity. As was noted by Emerton in a comment above, this answer to a related question shows that the answer is no, for an injective R-module I, the sheaf widetildeI is not necessarily an injective sheaf.
So if you like that answer, I suggest you click on the link above and upvote that answer.
The reference provided in that answer is to the following:
MR0617087 (82i:13013)
Dade, Everett C.
Localization of injective modules.
J. Algebra 69 (1981), no. 2, 416--425.
Localization of modules over a commutative ring R with respect to a multiplicatively closed subset S of R is an exact functor with a large number of properties, some of which are listed in Theorem 3.76 of J. J. Rotman's book [An introduction to homological algebra, Academic Press, New York, 1979; MR0538169 (80k:18001)]. The fifth property, namely: (LI) the localization S−1E of any injective R-module E is an injective S−1R-module, is false. Two examples are given here showing that arbitrary R and S need not have the property (LI). Also a positive result is given, showing that (LI) holds for certain non-Noetherian R and certain S. In particular, if R is the polynomial ring k[x1,x2,cdots] in a countable number of xn over a nonzero Noetherian ring k, then (LI) holds for all choices of S.
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