Maybe you mean "$\sigma$ is the same in all directions" -- a spherically symmetric distribution. That would force the above density, and independence of the coordinates $x_i$.
If you mean the joint normal distribution of several coordinate "dimensions" $X_i$, where each dimension considered by itself is normal with the same $\sigma$, then the formula holds if and only if all these dimensions are statistically independent of each other (no correlation between different coordinates).
Edit: ... that is all true if you remove the integral signs from the formula. For the formula as it was posted, $f(x)$ is independent of $x$ because of the integration.