Direct American aid to carry out abortions was already prohibited, but the rule Mr. Trump revived goes further. It requires not only hospitals or clinics to refrain from providing advice or information on the benefits and availability of abortion. It also requires any international organizations supporting those clinics to stop promoting abortion or advocating abortion rights anywhere in the world — even if they use non-American money to do so.
Experts say the rule has cut American aid to groups offering a wide range of services, not just abortion, during previous Republican administrations that have adopted the policy.
But this time, they say, the impact could be much bigger.
The wording in the Trump order extends the restrictions to all American global health aid, an $8.5 billion pot of money, according to an analysis by the Kaiser Family Foundation, a research organization. More than half of that money goes to programs for H.I.V. and AIDS, including services for women of reproductive age, the analysis found. An additional 9 percent goes to maternal and child health care, which is partly aimed at promoting safe pregnancies.
By contrast, the last time the rule was in place, under President George W. Bush, it applied only to family planning money, an amount that is currently around $520 million, the analysis found.
As health providers braced for cuts from Washington, the Netherlands lost no time in casting itself as a defender of reproductive rights. Its Foreign Ministry said Wednesday that it would start an international fund “to make up as much as possible for this financial blow.” It gave no specifics.
A spokesman for U.S.A.I.D. said the agency was still reviewing Mr. Trump’s presidential order to figure out how it would be enacted. But the language suggesting that it would be extended to all global health aid quickly sent a chill through the network of health providers that rely on American assistance to deliver a wide range of services in countries with fragile health care systems.
In Swaziland, which has one of the world’s highest H.I.V. infection rates, one private health care provider said its staff members provide abortion information to H.I.V.-infected women when necessary. Swaziland allows abortion only in cases of rape and incest, but patients can be referred to neighboring South Africa, where abortion is legal.
“Our organization could definitely be affected, including our H.I.V. services, and you can imagine how detrimental that could be for a small country like Swaziland that’s been heavily affected by H.I.V.,” said Zelda Nhlabatsi, the executive director of the Family Life Association of Swaziland, which says it receives a quarter of its funding from the American government.
In Lesotho, a landlocked nation surrounded by South Africa, Lerotholi Pheko, the executive director of the Lesotho Planned Parenthood Association, said he feared a hit to his operating budget even though his clinic does not provide abortion counseling.
The reason: His clinic receives most of its funding from the International Planned Parenthood Federation, which has said that it will lose $100 million in American funding over the next four years under Mr. Trump’s order.
“If we are not able to increase the income we get locally,” Mr. Pheko said, “it would mean that we would have to downsize.”
Here in Senegal, the small waiting room at the clinic on the busy University of Dakar campus was crammed with young women this week. Some cradled smartphones and tablets with headphone cords dangling from their ears.
Students in their early 20s arrived with questions and confusion about options for pregnancy prevention in a nation where talking about contraception is still largely taboo. A box of condoms was on a shelf, and students can be tested for sexually transmitted diseases.
“We are your brothers and sisters who listen without judging,” read a banner at the center.
Ms. Lancelot said the clinic does not provide abortion services or counseling. But the organization that runs it, Marie Stopes International, based in London, provides abortions and abortion counseling in other countries where it is legal, though not with money from the United States.
Marjorie Newman-Williams, director of operations for Marie Stopes International, which received $30 million in American funding 2016, said her group could not accept the Trump administration’s restrictions and would seek aid elsewhere.
“There will be a huge void in service delivery, and unless we can make up that money really fast, the funding won’t be there,” Ms. Newman-Williams said.
President Trump’s decision drew support from opponents of abortion in the United States. Jim DeMint, the president of the Heritage Foundation, issued his congratulations to Mr. Trump on Twitter “for reinstating Reagan’s #MexicoCityPolicy, preventing our tax dollars from funding abortion overseas.”
The Mexico City policy has been in effect under every Republican administration since President Ronald Reagan announced it in 1984. And it has been revoked by every Democratic administration.
The international groups affected by the prohibition usually provide a range of women’s health services, including pregnancy tests, H.I.V. tests and screenings for sexually transmitted diseases. To continue to receive American aid, the groups would have to certify that they do not offer abortion counseling, refer patients to abortion services or advocate legal abortions in the countries where they work. The policy spells out some exceptions, including cases of rape and treating women who have had botched abortions.
In Bogotá, Colombia, Marta Royo, who leads an organization called Profamilia, said she found the restrictions unacceptable.
Colombia legalized abortion in 2006, so to be barred from talking about it with patients, Ms. Royo said, would be “going against the country rules and the rights of the country that we have fought for.”
“It would be such a contradiction,” she said.
Her group, which is part of the International Planned Parenthood Federation, does not use American funding for abortion services, she said, only for family planning.
Research suggests that the policy has had a counterintuitive impact in the past. In countries that relied heavily on funding from the United States for reproductive health services, abortion rates rose when the Reagan-era policy was in place.
Researchers cite a possible reason: The aid spigot dries up for the organizations that provide contraceptive services to prevent unwanted pregnancies.
Avoiding such unintended pregnancies is what brought many of the young college students to the health clinic in Dakar. One 24-year-old student, Absa, who withheld her last name because her parents did not know she was sexually active, said she came to the clinic to ask about birth control after hearing a debate about contraception on television.
Another, Raicha, who was also worried about being condemned by her parents, came to the clinic after her boyfriend refused to use a condom. She worried she would get pregnant, so the couple agreed to look into using other contraceptives.
“I don’t want to get pregnant,” said Raicha, 22. “I want to finish my studies and be free to work.”