[CuSO4]

Cu²⁺ + 2H₂O  Cu(OH)₂ + 2H₃O+

I know that CuSO₄ is acidic, but it doesn't have any hydrogen atoms in it, so this is the only way I can see it making a acidic solution.

My thinking is that since Cu(OH)₂ is weakly basic, this reaction also can go the other way and be a acid too.

[Arkcon]

No.  This sort of question is popular lately.  An acid has have a proton to donate only if you use the traditional Arrhenius definition of acid:  http://en.wikipedia.org/wiki/Acid-base_reaction#Arrhenius_definition  But many substances raise or lower pH without ionizing into H⁺ or OH^-.  Some change pH by pulling something away, that is acting as a Bronsted-Lowery or Lewis acid or base. http://en.wikipedia.org/wiki/Br%C3%B8nsted%E2%80%93Lowry_acid%E2%80%93base_theory#Bronsted_concept_and_Lewis_acids.2Fbases.  This is how transition metal slats have a slightly acidic pH.

[Big-Daddy]

My thinking is that since Cu(OH)₂ is weakly basic, this reaction also can go the other way and be a acid too.

That's true but this makes Cu(OH)₂ a (weak) base, not Cu²⁺ - which is still an acid, because its hydrolysis leads to production of protons.

[CuSO4]

I meant to say that Cu²⁺ is slightly acidic not Cu(OH)₂. So the reaction i wrote is correct?

[suzi9spal]

This is how my professor explained acidity of salts to me:
CuSO₄ is a salt formed when combining Cu(OH)₂ and H₂SO₄.
Cu(OH)₂ + H₂SO₄  CuSO₄ + 2H₂O
If you look at the two reactants, Copper(II) hydroxide is a weak base, but Sulfuric(VI) acid is a really strong acid. Because of that difference, a salt of those two would be acidic.
Example: kitchen salt(NaCl) is a salt of sodium hydroxide, a strong base, and hydrochloric acid, which is a strong acid. That makes kitchen salt neutral.
baking soda(NaHCO₃) is a salt of sodium hydroxide, again a strong base, and carbonic acid, which is weak. That makes baking soda basic.
None of the alkali metals have acidic salts, because alkali metal hydroxides are very strong bases. Most transition metal salts are acidic though, because transition metal hydroxides are relatively weak bases.