Can relativization results be used to prove sentences formally independent?
There are no "natural" complexity-theory questions that have been proved independent of really powerful formal systems, such as ZF set theory or Peano Arithmetic.  (One could certainly construct such a question artificially, by playing games with G&ouml;del sentences.)

On the other hand, yes, you can interpret the statement that a sentence S relativizes as meaning that S can be proved from a certain restricted set of axioms (basically, the "Cobham axioms" that characterize closure under polynomial-time reductions).  Conversely, the existence of oracles making S either true or false is equivalent to S being independent of those particular axioms.  Here's the paper to read about this, by Arora, Impagliazzo, and Vazirani.

This is a very pretty connection mathematically---but it's worth stressing that we do have techniques (such as arithmetization) that go outside the relativizing axioms.  And I don't know of any results of the form "if natural open problem P can be solved at all, then it can also be solved in a relativizing way."