Is water acidic enough for an redox reaction involving H+?
Yes the reaction would be spontaneous at standard conditions.  However standard conditions for electrochemistry are at pH 0, i.e. an $[\ce{H+}]$ concentration of 1 molar.  

Solutions of lead nitrate by themselves are not going to give you pH 0.  Apparently very very concentrated lead nitrate solutions will reach pH 3 or 4, due to the weakly acidic nature of hydrated lead ions.

The electrochemical potential of the reaction you are interested in at pH 3 can be calculated from the standard (i.e. pH 0) potential via the Nernst equation.

$$E_\mathrm{cathode} = E_0\;– \frac{RT}{nF}\ln{Q}$$

Here, $Q$ is the "half"-reaction quotient, given by
$$ Q = \frac{\mathrm{p}_{\ce{N2O4}}}{[\ce{NO3-}]^2[\ce{H+}]^4}$$ Now assume that nitrate concentration and the partial pressure of dinitrogen tetroxide are in their standard states.  

If you go through the math (see examples here or here), then you will see that there is a 118 millivolt-per-decade decrease in the potential, i.e. every increase in the pH by 1 unit decreases the potential by 118 mV.  So already at pH 3 the reaction will no longer be favorable.  

Copper is unstable to concentrated nitric acid, but the acid needs to be concentrated.  Dilute acids cannot oxidize copper so easily.