Lightroom is a non-destructive editor, to complete the task you are trying to do you would need to export the image to a pixel editor, like Photoshop, where it would be displayed on a canvas. Lightroom doesn't have a canvas as it doesn't use layers.
Once exported, say to Photoshop, you could use a clone tool to fill the empty spaces in the canvas.
Or you could export as tif, psd or jpg and import to LR and use the clone tool.
Unfortunately that is exactly what I need to do: first export to a dummy, than re-import in Lightroom and do the simple editing.
Isn't it a worthless and time-consuming procedure?
I would love the ability of painting over those white areas directly in Lightroom as part of my workflow.
I guess that is why they call it 'non-destructive'