Neither. 'Ground' as you are using it is uncountable but it's only really used to describe the surface outside. Inside we say 'the floor'. So 'cover the whole floor'. This is a bit ambiguous because a carpet could also 'cover the whole floor' but if the context is clear, it'll be fine. 'Grounds' (plural) could be coffee beans which have been ground, reasons for thinking or found something, stadiums, but not the surface we walk on.