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When hydrogen peroxide acts as an oxidizer it itself is reduced. Broadly this means that the peroxide bond is broken and that water is formed.

However, why is there a tendency for hydrogen peroxide to be such a potent oxidizer in acidic media?

In acidic solutions, H2O2 is one of the most powerful oxidizers known—stronger than chlorine, chlorine dioxide, and potassium permanganate.

Why is it not as potent without added acid, i.e. 3% hydrogen peroxide, the stuff you find in medicine cabinets, isn't a very good oxidizer?

The "duh" answer is that hydrogen peroxide is especially unstable in acidic media. But why can't acidic media stabilize hydrogen peroxide, and specifically the peroxide bond?

Simple electrostatics tells me that having a bunch of positively charged hydronium ions around the positively polarized oxygens in the peroxide bond isn't a "good" situation. Of course there are anions from the dissociation of the strong acid but these anions necessarily have diffuse negative charge, which is probably not enough to stabilize the polarized peroxide bond to significant extent. Is there anything else?

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Just consider the Nernst equation for the half-reaction $$\ce{H2O2 + 2e- -> 2OH-}$$

You see the potential is clearly pH-dependent. Increased acidity will decrease the concentration of the reduced form, thus shifting the equilibrium to the right and consequently making peroxide a more potent oxidizer.

(BTW, even pH-neutral hydrogen peroxide is far from harmless. It can leave white marks on your hands and gnaw a hole in your jeans, albeit not instantly.)

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The immediate idea that jumps to mind is that acids, especially concentrated acids, have the ability to protonate the hydrogen peroxide. The then protonated hydrogen peroxide can then loose a water molecule either in an $\mathrm{S_N 2}$ type mechanism with rear-side attack on the other oxygen or in an $\mathrm{S_N 1}$ type mechanism to create a hydroxyl cation — although I would deem the latter highly unlikely due to the extreme electrophility of a potential hydroxy cation.

$$\ce{HO-OH + H+ -> HO-OH2+}$$

In the rear-attack version, this would directly void the attacking atom of two electrons formally (unless fluorine attacks which is even less likely). In the other case, the cation would probably jump to anything that is not oxygen and has electrons to share.

Additionally, hydrogen peroxide is stabilised by interactions of the free p-orbital of one oxygen and the $\sigma^*(\ce{OH})$ orbital on the other oxygen. (And vice-versa, of course.) When protonated, the proton will likely be added to the p-type lone pair rather than to the sp-hybrid type lone pair, inhibiting this stabilising interaction and further decreasing the strength of the $\ce{O-O}$ bond.

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