Only if you scale the border with the resolution and then keep it like that, which you wouldn’t do because it defeats the purpose of what you’re trying to achieve. You just increase the size of the image so that you have more pixels to work with, and then you reduce the width of the border until you’re satisfied. Look at the two buttons in the image below.
Their borders are the same width even though the one on the bottom is 4 times as large on either side. On way to achieve that result is to scale the smaller one up 4x and then divide the border width by 4 so that you end up with a border the same width as before, but on a larger image.
As an aside, I would strongly suggest to author UI backgrounds and icons as vector graphics because it allows for a more non-destructive workflow and scaling things becomes trivial while quality always stays at 100%.