In a statement, Mr Snowden also accused the US government of showing "no respect" for international law.
The US has charged Mr Snowden with leaking details of its electronic surveillance programmes.
Washington has expressed its "extreme disappointment" at Russia's decision.
White House spokesman Jay Carney said they were considering whether a meeting between US President Barack Obama and his Russian counterpart Vladimir Putin in September should go ahead.
The latest developments came amid fresh revelations from the cache of documents leaked by Mr Snowden.
Documents seen by the UK's Guardian newspaper appear to show the US government paid at least £100m ($150m) to the UK's GCHQ spy agency to secure access to and influence over Britain's intelligence gathering programmes. » | Thursday, August 01, 2013
