We obtain an optimal bound for a Gaussian approximation of a large class of vector-valued random processes. Our results provide a substantial generalization of earlier results that assume independence and/or stationarity. Based on the decay rate of the functional dependence measure, we quantify the error bound of the Gaussian approximation using the sample size $n$ and the moment condition. Under the assumption of $p$th finite moment, with $p>2$, this can range from a worst case rate of $n^{1/2}$ to the best case rate of $n^{1/p}$.