MOSCOW — Edward J. Snowden, the former National Security Agency contractor, fled the United States saying he did not want to live in a surveillance state.

But now the Russians are using his very presence here — on Friday Mr. Snowden said he intended to remain in Russia for some time while seeking asylum elsewhere — to push for tighter controls over the Internet.

Two members of Russia’s Parliament have cited Mr. Snowden’s leaks about N.S.A. spying as arguments to compel global Internet companies like Google and Microsoft to comply more closely with Russian rules on personal data storage.

These rules, rights groups say, might help safeguard personal data but also would open a back door for Russian law enforcement into services like Gmail.