The notion of white and colored noise
Yes. 

However, if you are dealing with a discrete-time signal, if your A/D is proceeded by an ideal low-pass filter with cutoff frequency $F_s/2$, with unity passband gain, the output power spectral density before sampling is:
$$
S_y(f) = |H(f)|^2 S_x(f) = \frac{N_0}{2} \operatorname{rect}(f/F_s)
$$
This means that the autocorrelation of the filtered noise will be:
$$
R_y(\tau) = \mathscr F^{-1}\{S_y(f)\} = \frac{N_0}{2T_s}\operatorname{sinc}(\tau/T_s)
$$
where $T_s = 1/F_s$. If the noise is zero-mean, Gaussian, and sampled at time instants $t_n = n T_s$, then each sample will be uncorrelated, and since they are Gaussian, will also be independent.   In this case, in the digital domain, the power spectrum will be flat for all frequencies. 

This is not the only way to generate noise that is white in the discrete-time domain, but is one way, albeit a physically unrealizable one.