No. Texture painting in 3D View is based on UVmapping.
There are many different ways to UVunwrap a mesh.
You can paint a texture that is not UVmapped in Image Editor.
If you want to paint an image in viewport that is not a shading texture but a modifier texture, UVs are also required.
Only other way to paint something in viewport is to paint attributes instead of textures.
But in that case, a special mapping would be ignored, too.