In this paper, we consider functionally indexed normalized weighted integral periodograms of equidistantly sampled multivariate continuous-time state-space models, which are multivariate continuous-time ARMA processes. With this, the sampling distance is fixed and the driving L\’evy process has at least a finite fourth-order moment. We derive the central limit theorem for the function-indexed normalized weighted integral periodogram under various assumptions about the function space and the moments of the driving L\’evy process. Either the assumption of the function space or the assumption of the existence of the moments of the L\’evy process is weak. Furthermore, we show weak convergence to Gaussian processes in both the space of continuous functions and the dual space, giving an explicit representation of the covariance function. The results can be used to derive the asymptotic behavior of the Whittle estimator and produce goodness-of-fit test statistics as Grenander-Rosenblatt and Cram\’er-von Mises statistics. We present the exact marginal distributions of both statistics and demonstrate their performance through simulation studies.