President Trump premised his entire campaign on the promise to “Make America Great Again.”

But on the international stage, many are asking if his actions threaten to do the opposite.

Martin Schulz, the former European Parliament president who is now vying to become Germany’s next chancellor, has used the term “un-American” to describe Mr. Trump’s policies. So has Mary Robinson, the former president of Ireland and recipient of the US Presidential Medal of Freedom. The Guardian newspaper invoked it in reacting to Trump’s travel ban.

Acting or being “American” has long had dual, contradictory meanings. With the United States’ status as sole superpower of the world since the fall of Communism, “America” has often carried a negative connotation of bullying and bellicosity.

Yet Mr. Schulz was referring to the other image, he told the German media group Funke recently: the America that captures the imagination and the dreams of people around the world when it comes to enlightenment, democracy, and freedom.

That is what many have in mind as they call to safeguard American ideals, and as protests have gained steam in the US and abroad over Trump’s orders to bar citizens from seven Muslim-majority countries for 90 days, suspend the US refugee program for 120 days, bar Syrian refugees indefinitely, and build a wall at the Mexican border. Suddenly the words of George Washington, who welcomed the “oppressed and persecuted of all nations and religions” to “the bosom of America,” are being brought up by those who have long been the US’s closest allies.

For all its perceived arrogance that has so often enraged the globe – moving unilaterally on the 2003 invasion of Iraq, engaging in drone warfare despite an international outcry, supporting dictatorships in the interests of stability or the economy, and asserting its commitment to human rights abroad despite failures in improving race relations and equality at home – the US has always stood as a beacon that many in the world want to see shine.

“All over the world, people lived in the hope and comfort of knowing that however badly things went in their countries, or elsewhere in the world, there was always this place called America,” says Simon Anholt, an independent policy adviser who founded the Good Country Index, which ranks countries on their contributions to the common good.

Stephane Madilian, a cobbler in his 40s in Paris, pauses when asked what the word “America” means to him. “It is grandeur, freedom, music, cinema; it’s an example. It’s the country that makes you dream.”

For Abdul Musa, a grocery store owner in Johannesburg, South Africa, who grew up during the country’s transition from minority white rule to democracy, America has meant possibility. “There was this place where democracy had survived for so many years and where it seemed like you could make it, whatever color you were,” he says.

And Zhang Yichi, an interpreter in Beijing in her late 20s, says her earliest memories of the US are from Hollywood movies. But when asked to define what the country most stands for, she chooses the word “democracy.”

Now she’s tacked on two more: “Democracy in chaos.”

How the US ranks in indexes

The world has depended on the US to lead the democratic liberal order since the end of World War II, from putting its military muscle behind NATO to ensuring rules-based and open trade under the World Trade Organization. The US has poured its vast wealth, sometimes with little thanks, into anti-corruption efforts around the globe, into the HIV/AIDS response in Africa, into recovery efforts after tragedies like the 2004 tsunami that was centered in Indonesia. This is partially why it sits at No. 1 on the Anholt-GfK Nation Brands Index, which measures global perceptions of countries – “the most admired country in the world,” Mr. Anholt says.

It is No. 20, however, on his Good Country Index, which weighs actions such as contributions to global peace and security or progress on climate change. It’s an indication of the competing narratives about the US, Anholt says.

“People’s views of the US can be complicated,” says Richard Wike, director of global attitudes research at the Pew Research Center. “There’s often a lot people admire about America,” he says, including its political ideals, pop culture, and progress in science and technology. “But they sometimes worry about American power, and how they use that power.”

The temporary travel ban that Trump put in place for citizens of Iraq, Iran, Syria, Somalia, Sudan, Yemen, and Libya is seen by many as using the administration’s power for bad. And perhaps no one is as conflicted as those directly affected, including one young Iranian professional who had been on course to enter a PhD program in Massachusetts – his first choice for a niche program in genocide and holocaust studies.

“Right now, that is vanishing into thin air,” said the husband and father, who asked to remain anonymous. He spoke from Tehran, Iran, before US District Judge James Robart blocked Trump’s ban across the US.

The ban is not just killing dreams. Members of the US State Department, addressing the issue through the department’s “dissent channel,” warned of potential anti-Americanism among a rising generation.

“Almost one-third of these countries’ combined populations are children under the age of 15; there is no question that their perception of the United States will be heavily colored by this ban,” the memo states. “We are directly impact[ing] the attitudes of current and future leaders in these societies – including those for whom this may be a tipping point towards radicalization.”

Of the seven nations listed on the ban, Iran could be the most deeply affected. Iranians joke about “Tehrangeles,” with Los Angeles just one of many places Iranians have moved to in large numbers since the 1979 Islamic revolution.

And despite media attention given to US flag burning, and Iran’s hard-liners chanting “Death to America” and calling the US the “Great Satan,” the country’s population is widely recognized as one of the most pro-American in the Middle East. It was in fact the notions of freedom, diversity, and the “American dream” that attracted the would-be PhD student to New England.

Even if Trump’s orders get overturned, the student questions how secure American principles ultimately are. “Without those values, America is not America; it’s just a simple country like other places in the world.”

A complicated US relationship with Mexico

The US can never be just any other place for Mexico, whose giant neighbor to the north has long loomed over it. “For Mexico, the world is often reduced to the United States,” says Lorenzo Meyer, a well-known Mexican historian.

Anti-Americanism has simmered in Mexico ever since it lost a large swath of territory to the US in 1848. Relations warmed at the outset of World War II and through the cold war, as the nations found themselves on the same side. And while the war on drugs has been fraught, economic convergence between the two, as well as mass immigration, has served as a powerful antidote to nationalism.

The North American Free Trade Agreement, which Trump has promised to rewrite, was the real turning point in the 1990s. “There was a qualitative change in Mexico: Mexico stopped seeing the US as a threat and began to see it as an opportunity,” says Héctor Aguilar Camín, founder of the cultural and political magazine Nexos.

Now with Trump promising to move forward to erect a wall and make Mexico pay for it, some worry that nationalism could make a comeback – as latent suspicions already have.

“Deep down, history has taught Mexico that at any moment the US could convert into a problem, and we’re seeing that today,” says Mr. Meyer.

This is not the first time other countries have had a problem with a US president. Approval of the US plummeted during George W. Bush’s administration, and opinions are volatile. In Germany, a Pew survey showed a swing in approval for the US president from 14 percent in 2008 (Mr. Bush’s last year in office) to 93 percent in 2009 (Barack Obama’s first). Approval of the US went from 31 percent to 64 percent in the same time period.

Gloria de Rivera, a hairdresser in Mexico City, says she’s not worried about permanent damage to the US-Mexican dynamic because of the power of money.

“The US is a place of employment,” she says. “No matter what Trump does or says, the US relies on hardworking Mexicans.” And Mexicans depend on the US as an economic escape valve. That, she says, won’t change.

Worries in Africa

But others are far less optimistic about the ramifications. Many across the African continent have seen the country as a beacon, especially after Americans elected their first African-American president.

But Ralph Mathekga, a South African political analyst, says Africa’s 54 countries have so far failed to figure in Trump’s policy. “Trump’s views are becoming very clear: Anything outside the Western world is just a little tin-pot country not worth his time,” Mr. Mathekga says. “We have reason to worry, I think, that this sort of racism towards our part of the world can become institutionalized – in trade policies, in foreign policies, in how aid is distributed.”

Beyond individual countries or regions, many worry that a new America could upend the old order. Departing from prepared remarks during her final speech as chairperson of the African Union, South African diplomat Nkosazana Dlamini-Zuma said recently, “It is clear that globally, we are entering very turbulent times.”

Huang Jing, a professor at the National University of Singapore who focuses on US-China relations, became a naturalized US citizen in 1999 while teaching at the University of Utah. Since then he has had conflicted feelings about his second home – he was especially dismayed by Bush’s foreign policy – but he has always maintained faith in US institutions and norms. While he still holds onto that faith, Mr. Huang says his worry is mounting.

“The damage is not only done to the United States,” he says, “but to the entire liberal order of the world that the US has helped maintain and lead for the last 70 years,” he says. “Not only do we not have a lighthouse, but we also don’t have a policeman on earth anymore.”

Pressures in Europe

Satirical videos have popped up across Europe, starting in the Netherlands, mocking Trump’s “America First” take on foreign policy. But many aren’t laughing about the real-world politics they are witnessing. Trump’s administration is particularly threatening to Europe because it reflects sentiments that are driving political change at home, says Rebecca Harms, a German deputy for the Greens in the European Parliament. “All of these ... old political ideas that the nations are great only if they are on their own have made a comeback, not only in the US but in the EU,” she says.

The European Union, especially after Britain’s vote to leave the bloc, faces growing pressures, from the surge of refugees to a virulent populism.

“Especially for security reasons, the EU has to check its own capacities and has to be prepared for a major change in relations,” says Ms. Harms.

The growth of populist parties across Europe, from the far-right National Front in France to the Freedom Party of Geert Wilders in the Netherlands, has Europe bracing for disruption. And now the continent may no longer be able to turn to the US to maintain order. Even before Trump’s comments that NATO was “obsolete,” the EU had been boosting expenditures on defense and fostering more cooperation.

Still, Europeans like Mr. Madilian, the cobbler, worry that the influence of other powers like Russia might grow in the vacuum that an “America First” policy might generate.

Against this backdrop, watching the changes in the US is unsettling, says Madilian, as he repairs shoe soles on a recent afternoon.

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“One advantage [the US had] is that whoever won the elections, whoever was in power, America stayed America – everybody got behind the president,” he says. “Now I don’t know what is going to happen.”

• Whitney Eulich in Mexico City; Michael Holtz in Beijing; Peter Ford in Paris; Scott Peterson in Istanbul, Turkey; and Ryan Lenora Brown in Johannesburg, South Africa, contributed to this report.