I don't understand what covariance matrix you're talking about. Let's say we have two variables and the joint distribution is given by the probability density

I don't understand what covariance matrix you're talking about. Let's say we have two variables and the joint distribution is given by the probability density function p(x,y) which can have any form as long as it's always >=0 and integrates to 1. The covariance matrix is a 2x2 matrix and contains only three real numbers (four, but the off-diagonal terms are necessarily equal). The covariance matrix (together with the vector of means) gives a complete description of the joint distribution only when the joint distribution is completely specified by its first two moments and all its higher moments are zero (i.e. it is a multivariate normal). The fact that the correlation between the variables is zero doesn't mean necessarily that the variables are independent when the joint distribution is not a multivariate normal. The covariance matrix alone cannot be used to tell whether the joint distribution is a multivariate normal or not. Every possible covariance matrix defines a multivariate normal but is also compatible with an infinity of other distributions.