Why is the inside of a cell both more acidic and more negative than its environment? In a very simplistic way of thinking, an acidic solution has more positively charged $\ce{H+}$ ions present, whereas a basic solution has more negatively charged $\ce{OH-}$ ions present.

Can that be simply reconciled with the fact that (1) the intracellular pH of about 6.8 is more acidic than the intercellular pH of about 7.3, and that (2) a typical cellular membrane potential is -40 to -80 mV?
The key is that $\ce{H+}$ and $\ce{OH−}$ are not the only ions at play here.  

The selective permeability of different cellular membranes for different anions (i.e. $\ce{Cl−}$) and cations (i.e. $\ce{K+, Na+, Ca^{2+}, Mg^{2+},}$ etc.) results in charge separation. Cellular membrane potentials are thereby created, with or without an intercellular / intracellular pH gradient.