If you make the statement

Fix an algebraically closed base field k and let X be a scheme of finite type over k. Then 
X/k is proper iff for all smooth quasi-projective curves C/k and all maps f: C\c -> X then f extends uniquely to f': C -> X.

it seems true to me. This should basically comes down to the fact that one can classify birational equivalence classes of curves over k in terms of the 'abstract curves' coming from all possible discrete valuations on dimension 1 function fields K/k. So using the fact that in this situation it is sufficient to check the valuative criterion on DVRs it seems like it should not be so hard to see the equivalence.

For the same reasons this should work for checking separatedness when one makes the obvious modifications to the statement.

Over other bases I am not sure at the moment... I can't remember if the birational classification is still that simple (although some people implicitly mean by variety that everything is over some fixed alg. closed base).

In the more general case (if your definition of variety doesn't include a finite type over a noetherian base hypothesis) where one needs non-noetherian valuation rings I think this interpretation is false - non-noetherian valuation rings can have arbitrary dimension.