"Behind this is unfriendly competition, banal corruption and an inhumane policy in which diseases, the suffering and death of people are secondary," Naryshkin said, according to the statement published by the SWR security service.
He said the EU was preventing the approval of the Russian vaccine Sputnik V in Europe for political reasons.
The European Medicines Agency (EMA) has been reviewing Sputnik V for weeks and is expected to reach a decision on its approval in the summer.
"The resistance of EU officials is a real brake on the expansion of the scale of vaccinations in European countries," Naryshkin said.
Russia has repeatedly accused the West of political manoeuvring against the vaccine, which has now been approved for use in nearly 70 countries.
However, Russian leaders also said early on in the pandemic that the country would not approve widely used Western vaccines made by companies such as Moderna and BioNTech, saying Russians should instead use vaccines developed in their own country instead.
Naryshkin also said the EU was preparing "another smear campaign" accusing Russia of supporting anti-vaccination movements abroad.
He rejected Western accusations of Russian activities abroad, in an interview by the BBC broadcaster.
"These issues related to poisonings, cyberattacks and hacking, interference in elections attributed to Russia are so abstruse and sometimes even poor," he said, likening them to a "bad detective story."
He also said that the SWR is prepared for talks with British external intelligence agency MI6, and had written to agency chief Richard Moore and hoped to hear back from him.
Russian government spokesperson Dmitry Peskov welcomed the initiative, saying such contact was in the two countries' mutual interest, according to Interfax news agency.
Russia's relations with Britain are currently at a low point, as they are with the EU and the United States, with points of contention ranging from the attempted poisonings of dissidents and former spies, to allegations of election interference and the annexation of Crimea, among other issues.