"On the basis of what we have learnt in the Salisbury investigation and what we know about this organisation more broadly, we must now step up our collective efforts specifically against the GRU," she said.
Two Russian nationals suspected of the attempted murder of former Russian spy Sergei Skripal and his daughter Yulia, using the names Alexander Petrov and Ruslan Boshirov, are thought to be officers from Russia's military intelligence service, the PM said.
Scotland Yard and the Crown Prosecution Service (CPS) say there is enough evidence to charge the men.
The two suspects, believed to have been using aliases, travelled on Russian passports and are thought to be about 40.
The CPS is not applying to Russia for the extradition of the two men, as Russia does not extradite its own nationals.
But a European Arrest Warrant has been obtained in case they travel to the EU.
In response, the Russian foreign ministry has said the names and photographs of the men "do not mean anything to Moscow".
Video by Sky News
Russian ex-spy Sergei Skripal, 66, and daughter Yulia Skripal, 33, remain critically ill in hospital, after they were found unconscious on a bench in the Wiltshire city on March 4. The UK government says they were poisoned with a nerve agent of a type developed by Russia called Novichok and PM Theresa May said she believed Moscow was "culpable."
Det Sgt Nick Bailey also fell ill after responding to the incident in Salisbury.
Police are linking the attack to a separate poisoning on 30 June, when Dawn Sturgess and Charlie Rowley became unwell at a house in Amesbury, about eight miles from Salisbury.
Ms Sturgess died in hospital on 9 July. Mr Rowley was discharged from hospital on 20 July.