Stellar core collapse events are expected to produce gravitational waves via several mechanisms, most of which are not yet fully understood due to the current limitations in the numerical simulations of these events. In this paper, we begin with an empirical functional form that fits the gravitational-wave spectra from existing simulations of stellar core collapse and integrate over all collapse events in the Universe to estimate the resulting stochastic gravitational-wave background. We then use a Gaussian functional form to separately fit and model a low-frequency peak in the core-collapse strain spectra, which likely occurs due to prompt convection. We systematically study the parameter space of both models, as well as the combined case, and investigate their detectability by upcoming gravitational-wave detectors, such as Advanced LIGO and the Einstein Telescope. Assuming realistic formation rates for progenitors of core-collapse supernovae, our results indicate that both models are 2-4 orders of magnitude below the expected sensitivity of Advanced LIGO, and 1-2 orders of magnitude below that of the Einstein Telescope.