There is no way to adjust line width, the API only provides rendering of 1-pixel-wide lines. If you want something else you'll have to roll your own solution. One possible approach might be to use a sprite renderer to draw your lines.

In general D3D11 doesn't expose features that aren't supported by the hardware, which is surely why there is no option for line width. I'm no OpenGL expert, but a quick search seems to indicate that line widths > 1.0 aren't supported in the core profile since version 3.2.

One way to generalize lines on GPU's is by rendering quads with width and height determined based on the thickness and length of the line you want to render. A sprite renderer will essentially render and batch quads, which is why I suggested using one and linked you to DirectXTK. A sprite renderer will also let you apply a texture, which can let you approximate antialiasing as well as different line styles. Another possible quad-based approach is to render a quad that covers the the screen-space area covered by a line, and then use a pixel shader to analytically determine coverage and rasterize the line. This article explains how to do this.

A geometry shader is generally used for expansion from one primitive to multiple primitives. An example would be taking a single point/vertex as input, and emitting a quad as an output. This is done frequently for particle systems, and could also be used in a sprite or line renderer to expand points to quads. You could also take a line primitive and emit a quad as a means implementing line thickness.