Your description of the origin of Hawking radiation is the one commonly used in popular science articles, but it just a metaphor and not really what happens. The vacuum is not full of virtual particles popping into existence and disappearing again. For more on this see Are vacuum fluctuations really happening all the time?
The actual reason for Hawking radiation is rather more complicated than this. There is an attempt to explain it in relatively simple terms in An explanation of Hawking Radiation but even this will be difficult for non-physicists. However the key point you need to take away is that no Hawking radiation exists unless there is an event horizon present. That means the universe cannot create Hawking radiation just because spacetime is expanding.
However if the expansion is accelerating then we do get an event horizon around the visible universe called the cosmological event horizon, and this event horizon does produce Hawking radiation. And indeed it seems that the expansion is indeed accelerating due to the presence of dark energy. This means that as I write this a cosmological event horizon is forming around us, though technically it will take an infinite time to form. The Hawking radiation produced by the cosmological event horizon is far too small ever to be measured, but it does exist.