Why is the inside of a cell both more acidic and more negative than its environment?
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.