Inverse Discrete Time Fourier Transform of a given Function using the definition
Yes, you can do that. Because we substitute $e^{j\omega}$ for $z$ to get the frequency response of a discrete system, you can go back by substituting $z$ for $e^{j\omega}$. This yields

$$H(z) = \frac{1}{(2-z)(2-z^{-1})} = \frac{1}{5-2z^{-1}-2z} = \frac{z^{-1}}{-2+5z^{-1}-2z^{-2}}$$

Therefore the system you're trying to find is

$$-2y[n]+5y[n-1]-2y[n-2] = -x[n]$$

$$y[n] = \frac{-x[n]-5y[n-1]+2y[n-2]}{-2} = \frac{1}{2}x[n] + \frac{5}{2}y[n-1] - y[n-2]$$