I asked the same question when I took my first organic chemistry course. Actually, the H IS very slightly acidic and it IS very slightly basic. The pKa as an acid is about 16-19 which makes it very much weaker than acetic acid (vinegar) with a pKa of about 5 but still acid enough to form the alkoxides with sodium metal or NaH; i.e., CH3COOH + Na ==> CH3COONa + H2.
The oxygen of the OH group also has two sets of unpaired electrons which makes it usable as a base albeit a very weak base. For example, in the presence of very strong acids, such as H2SO4, it can form the CH3COOH2^+ which makes it a base because it has accepted a proton. Take a look at this site and scroll down to "physical and chemical properties" where they have good diagrams to show this action.
http://en.wikipedia.org/wiki/Alcohol
For the alcohol functional group, why is the "H" not acidic and the "OH" not basic?
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