Im a beginner in signal processing so my question may be obvious.

A white noise has the property to have its autocorrelation function that is equal to 

$$\mathbb{E}[f(t+\tau)f(t)]=\sigma^2 \delta(\tau).$$

We say that random signals can be Gaussian white noise, but for me if the signal has a Gaussian distribution it is necesseraly a white noise because as soon as we know the distribution then $\mathbb{E}[f(t+\tau)f(t)]$ is automatically determined.

Am I wrong ?