I understand that the larger your resolution, the more the size of your video memory matters. But, what if the display size in general is larger? For example, if I'm running two displays at the same resolution, but one is a 48 in. tv and the other is a 22 in. monitor, will the 48 in. screen benefit from more memory just because the screen is larger?
The size does not matter,mate. The pixel does. If the pixel is fixed by some number, the job left is to stretch it, which is often the case that confuses people.
To employ properly of a 1080p field of display, GPU needs to have an at least 1GB memory. And I'm not even starting on multi-display setup.
So the answer for your question is, it depend on the pixel resolution your display pull off. If it do, say 1920x1080 or so, then you'll most likely needed an extra vram on GPU, maybe somewhere around 1,5-2GB.
But get this too, despite the fact that vram size does matter on higher res, you must also consider the type of memory the said GPU employs. Typically, a high-end GPU uses a GDDR5, for such said resolution you'd still need at least 2GB anyway...