WASHINGTON -- Secretary of State John Kerry landed in Rome and Paris to talk about Mideast peace, Syria and Iran but was confronted by outrage over the sweep and scope of U.S. spying abroad. President Barack Obama already has defended America's surveillance activities to leaders of Russia, Mexico, Brazil, France and Germany and was even questioned about it during his birthday appearance on late-night television.

Obama and Kerry are trying to calm international anger over classified disclosures by former National Security Agency contractor Edward Snowden. Snowden's revelations about NSA tactics -- that allegedly include tapping as many as 35 world leaders' cellphones -- threaten to harm U.S. foreign policy in several areas.

In Washington, demonstrators held up signs reading "Thank you, Edward Snowden!" as they marched near the U.S. Capitol to demand that Congress investigate the NSA's mass surveillance programs.

"The magnitude of the eavesdropping is what shocked us," former French Foreign Minister Bernard Kouchner said in a radio interview. "Let's be honest, we eavesdrop too. Everyone is listening to everyone else. But we don't have the same means as the United States, which makes us jealous."

The British ambassador to Lebanon, Tom Fletcher, tweeted this week: "I work on assumption that 6+ countries tap my phone. Increasingly rare that diplomats say anything sensitive on calls."

Diplomatic relations are built on trust. If America's credibility is in question, the U.S. will find it harder to maintain alliances, influence world opinion and seal trade deals.

Spying among allies is not new. Madeleine Albright, secretary of state during the Clinton administration, recalled being at the United Nations and having the French ambassador ask her why she said something in a private conversation that the French had apparently intercepted. The French government protested revelations this week that the NSA had collected 70.3 million French-based telephone and electronic message records in a 30-day period.

Albright said Snowden's disclosures have been very damaging to U.S. policymakers.

"I think it has made life very difficult for Secretary Kerry," Albright said at a conference hosted by the Center for American Progress in Washington.

The disclosures could give the Europeans leverage in talks with the U.S. on a free trade agreement, which would bring together nearly half of the global economy. "If we go to the negotiations and we have the feeling those people with whom we negotiate know everything that we want to deal with in advance, how can we trust each other?" Martin Schulz, president of the European Parliament, asked.

To Henry Farrell and Martha Finnemore at George Washington University, damage from the NSA disclosures could "undermine Washington's ability to act hypocritically and get away with it."

Writing in the magazine Foreign Affairs, they claim the disclosures forced Washington to abandon its "naming-and-shaming campaign against Chinese hacking."

The revelations could undercut Washington's effort to fight terrorism, says Kiron Skinner, director of the Center for International Relations and Politics at Carnegie Mellon University. The sweeping nature of NSA surveillance goes against the Obama administration's claim that much of U.S. espionage is carried out to combat terrorism, she says.

"Allied leaders will have no incentive to put their own militaries at risk if they cannot trust U.S. leadership," Skinner said.

The Obama administration's response to the outrage has been that the U.S. is gathering foreign intelligence of the type gathered by all nations and that it's necessary to protect the U.S. and its allies against security threats.

Most governments have not retaliated, but some countries are pushing back.

Germany and France are demanding that the Obama administration agree by year's end to new rules that could mean an end to reported American eavesdropping on foreign leaders, companies and innocent citizens.

Brazilian President Dilma Rousseff cancelled her official state visit to the White House. And Brazil says it is working with other countries to draft a United Nations General Assembly resolution that would guarantee people's privacy in electronic communications.

A European Parliament committee in Brussels approved sweeping data protection rules that would strengthen online privacy and outlaw the kind of data transfers the U.S. is using for its spying program.

European lawmakers have called for the suspension of an agreement that grants U.S. authorities access to bank data needed for terror-related investigations.

"We need trust among allies and partners," said German Chancellor Angela Merkel, whose cellphone was allegedly tapped by the NSA. "Such trust now has to be built anew."