The immigration agreement imposed on Mexico by Donald Trump under the threat of punitive tariffs is a victory for “hostage-taking” over international rules, a former head of the World Trade Organization (WTO) said on Saturday.

In the US, critics of the president began to answer his triumphalism over the deal. The House speaker, Nancy Pelosi, said Trump had “undermined America’s pre-eminent leadership role in the world by recklessly threatening” Mexico.

“Threats and temper tantrums are no way to negotiate foreign policy,” she said.

The New York Times reported that key concessions from Mexico on immigration had in fact been agreed for months.

On Saturday, the Mexican president, Andrés Manuel López Obrado told a rally in Tijuana:

“We’re celebrating yesterday’s important agreement because it was putting us in a very difficult situation – these tariffs because we would have had to impose the same measures on US products.”

Late on Friday, Trump announced that the US and Mexico had struck an accord to avert a tariff war. Under the deal, Mexico agreed to expand a contentious asylum program, known as Remain in Mexico, that critics say puts migrants in danger from criminal cartels. López Obrador’s government also committed to deploying security forces to stem the flow of migrants from Central America.

Trump had threatened to slap escalating tariffs of 5% on all Mexican goods from Monday if López Obrador did not do more to tighten his country’s borders.

“My reaction is it seems that hostage-taking works,” Pascal Lamy, director-general of the WTO from 2005 to 2013, told Reuters, saying Trump’s actions went against the spirit of diplomacy.

“If there’s a rule of law, it’s because people believe it’s better than the law of the jungle. And many people don’t like the law of the jungle because some are strong, some are weak, and they don’t want the strong to always step on the weak.”

The White House did not immediately respond to requests for comment, although Trump had spent the day trumpeting his victory on Twitter.

In the morning, he claimed there had been “much false reporting” of his deal “by the fake and corrupt news media, such as Comcast/NBC, CNN, New York Times and Washington Post”.

He did not cite examples of reporting that had angered him, but in the afternoon the Times cited “officials from both countries who are familiar with the negotiations” when it said Mexico agreed the steps contained in the deal “months earlier” in meetings with former homeland security secretary Kirstjen Nielsen.

Trump, who spent the day at his golf course in Virginia, did not immediately respond to a report which also said it was unclear if he “believed that the agreement truly represented new and broader concessions, or whether the president understood the limits of the deal but accepted it as a face-saving way to escape from the political and economic consequences of imposing tariffs on Mexico”.

World markets have been roiled by Trump’s use of tariffs, fanning concern about the stability of multilateral institutions that grew up after the second world war. Lamy, a Frenchman, is a former European commissioner for trade. His criticism of Trump’s tariffs reflects wider misgivings. Trump has blamed the WTO for not doing enough to defend US interests, and in August 2018 threatened to pull out of the organization.

In her statement on Saturday, Pelosi lamented Trump’s behavior towards “our close friend and neighbor to the south” and said she was “deeply disappointed” by the “expansion of [the] failed Remain in Mexico policy, which violates the rights of asylum seekers under US law and fails to address the root causes of Central American migration.

“President Trump must stop sabotaging good-faith, constructive and bipartisan efforts in Congress to address this complex problem in a humane manner that honors and respects our most cherished national values,” she said.

“The Trump administration must also do much more to cooperate in a meaningful way with Mexico in cracking down on smuggling networks. It’s failure to do so thus far is unconscionable and irresponsible.

“Threats and temper tantrums are no way to negotiate foreign policy.”

Trump had earlier attacked her in a tweet, writing: “Nervous Nancy Pelosi [and] the Democrat House are getting nothing done.”

Mexico sends about 80% of its exports to the US, giving Trump ample leverage to put pressure on López Obrador over a surge in migrants to the US border.

Lamy said it was understandable that Mexico had sought to extricate itself from the tariff bind, but noted it ran the risk of facing more threats in future. He was in “absolutely no doubt” the WTO would have found in favor of Mexico if López Obrador had asked it to arbitrate the dispute with Trump, a process he said would have taken around two years.

“The US president is taking trade decisions that are in total violation of the WTO rules,” he said. “That was the case with these Mexican tariffs. The notion that you put a tariff because there are too many people crossing the border is just miles away from any letter and spirit of the WTO agreement. Which is why I qualify this as hostage-taking.”

Lamy said it was not clear whether Trump was interested in reforming the WTO or neutralizing it. He said Trump had complaints worth heeding, noting that some WTO rules made it hard to constrain Chinese trade practices that have caused frictions. But he said the rest of the world would need a fallback plan if the US decided not to abide by international rules.

“The others have to find a way to stabilize the multilateral rules-based system,” he said, “even if the US wants to kill it.”

Remarks from senior US officials suggested Trump had not abandoned the idea of threatening Mexico with tariffs.

In Japan, the treasury secretary, Steven Mnuchin, told reporters: “Our expectation is that Mexico will do what they’ve committed to do and our expectation is that we won’t need to put tariffs in place, but obviously if that’s not the case, the president retains that authority.”

The Times quoted an anonymous US official as saying: “The tariff threat is not gone. It’s suspended.”

Reuters and Victoria Bekiempis contributed to this report