With the advancements of computer architectures, the use of computational models proliferates to solve complex problems in many scientific applications such as nuclear physics and climate research. However, the potential of such models is often hindered because they tend to be computationally expensive and consequently ill-fitting for uncertainty quantification. Furthermore, they are usually not calibrated with real-time observations. We develop a computationally efficient algorithm based on variational Bayes inference (VBI) for calibration of computer models with Gaussian processes. Unfortunately, the speed and scalability of VBI diminishes when applied to the calibration framework with dependent data. To preserve the efficiency of VBI, we adopt a pairwise decomposition of the data likelihood using vine copulas that separate the information on dependence structure in data from their marginal distributions. We provide both theoretical and empirical evidence for the computational scalability of our methodology and describe all the necessary details for an efficient implementation of the proposed algorithm. We also demonstrate the opportunities given by our method for practitioners on a real data example through calibration of the Liquid Drop Model of nuclear binding energies.