Create a 1 pixel by 1 pixel white png image in paint. In Godot, place a sprite and set that png as its texture. In the inspector under CanvasItem/Visibility, set the field "Modulate" to your desired color. "Modulate" is unique to the node, so even if multiple sprites use the same png, they can be of different colors.
If you are intent on not using an image texture, then just select new GradientTexture on your sprite texture. Set the gradient width to 1 and the sizes of PoolFloatArray and PoolColorArray to 1 as well. Under PoolColorArray, select your color. Then just scale the sprite to the desired size.