Alison Gill, New York-based Human Rights Watch's Moscow chief, said: "We're allowed to go into our offices to pay our electricity bills, but we can't do anything concerning our work. We've suspended our work as of today."
The re-registration deadline expired on October 18 and involved almost impossible levels of paperwork.
Vladimir Putin, Russia's president, said the law was needed to combat terrorism and stop foreign spies using non-governmental organisations (NGOs) as cover.
Critics say it gives officials a free hand to crack down on civil society at a time when the Kremlin is exerting increasing control over the media and political parties.
Human Rights Watch, Amnesty International and the Danish Refugee Council, the only Western humanitarian agency working in the breakaway region of Chechnya, are among those forced to suspend work until their application to re-register is cleared.
Questions of human rights and press freedom were highlighted by the October 7 murder of journalist Anna Politkovskaya, who had created controversy with accusations of torture in Chechnya.
Lydia Aroyo, an Amnesty International spokeswoman in London, said: "Obviously we are not happy that we had to suspend our work. We are seeking clarification. We will continue researching human rights violations in Russia out of London."
The new law requires any foreign NGO operating in Russia to produce a large number of notarised documents, including passport numbers, home addresses and telephone numbers back in home countries, to a federal agency which decides whether to approve them.
Aroyo said: "The new law has no clear implementation guidelines. It is very cumbersome. It's given us lots of trouble."
Natalia Ronina, head of the unit dealing with NGOs at the Federal Registration Service, disagrees with the criticism.
She said: "The process is not very complicated. We have applications from many countries - Germany, France, Belgium, Ireland, Spain and the United States. We've had around 400 visitors."
Ronina said that as of Thursday, 95 foreign organisations had been approved, 95 were pending and five refused, whom she declined to name.
The law gives Russia the right to stop activities or close down groups it considers a threat to national security. Russian organisations are next on the list.
Lena Nemirovskaya, the founder and director of the Moscow School of Political Studies, a Russian think-tank, said: "Nobody knows what the new rules will be. This law is all about control."