Why does White Noise in images imply noise in adjacent pixels are independent?
How did the professor state this? Did they say that the noise is iid?  That means that the samples are independent, identically distributed.  That means they are independent by definition (they are assumed to be so).

White noise, while a little more vague, also makes the assumption of independence. Again, this means that the noise is assumed to be independent.

However, if all that is said is that the samples are uncorrelated with the given covariance matrix, then you are correct: this does not imply independence.