Why does a fully charged capacitor show infinite resistance?

The voltage-current relationship for a capacitor is i = C dv/dt, where i is the current flowing through, C is the capacitance and dv/dt is the time rate change of the voltage across that capacitor. So, when a capacitor is fully charged, the voltage no longer changes with time (the derivative, dv/dt, is now 0). As can be seen from the equation, the current would therefore be 0. Anything with 0 current flowing through is an open circuit, and can be treated like a resistor with infinite resistance (in models, anyway). Practically speaking, capacitors aren't this perfect, but you will still have an extremely high resistance once fully charged (voltage changes negligibly after charging).