In short, the US cannot functionally be independent on fossil fuels even if we extracted every drop of oil within our borders--because we literally cannot use all of it, and most of it would be wasted just sitting around.
Yes, we would need to build more refining capacity to use it all effectively, but in a cataclysmic event (i.e. a world war or something) the US would be much much better off than most other countries.
Europe, or China, or India could not though.
India has geography for solar, and the human/industrial capability for nuclear.
Southern Europe can go solar as well.
Northern Europe has it tougher (except Norway, with its abundant hydro). Nuclear could work. Or long range DC cables from South Europe or North Africa (if ever Europe helps them to put their act together - not easy or fast, but definitely in their best interest).
[1] https://en.wikipedia.org/wiki/Spain-Morocco_interconnection
Central and Eastern Europe have it tougher.
The Balkans are quite fossil-heavy, but solar should be quite feasible there.
If foreign oil supply were cut from China or India, they'd be in a much bigger trouble.