US President Donald Trump signed an executive order requiring the body of water – which is bordered by the US, Cuba and Mexico – be renamed the Gulf of America in his first week in office.

But it will only appear on Google Maps with the new name for people based in the US – elsewhere in the world it will retain its current name, which has been used for hundreds of years.

There is no international organisation responsible for the naming of bodies of water.

But Mexico argues the U.S. cannot legally change the Gulf’s name because the United Nations Convention on the Law of the Sea dictates that an individual country’s sovereign territory only extends up to 12 nautical miles out from the coastline.

“[The name change] could only correspond to the 12 nautical miles away from the coastlines of the United States of America,” Sheinbaum said.

Google has not yet responded to the BBC’s request for comment.

But in a statement on social media on Monday it said: “We have a longstanding practice of applying name changes when they have been updated in official government sources.”

It will also rename Mount Denali as Mount McKinley in the US, following another order from Trump.

“When official names vary between countries, Maps users see their official local name,” it said.

Sheinbaum has criticised Google’s decision, saying the firm should not respond to “the mandate of a country” to change the name of “an international sea”.

But she seemed to poke fun at Trump’s move by joking Mexico may demand Google make some additional renaming decisions.

“By the way, we are also going to ask for Mexican America to appear on the map,” she said.

Sheinbaum previously joked she would consider renaming North America as “América Mexicana” in the country.

“He says that he will call it the Gulf of America on its continental shelf,” Sheinbaum previously said when Trump signed the executive order.

“For us it is still the Gulf of Mexico, and for the entire world it is still the Gulf of Mexico.”

Among the numerous executive actions Trump signed in his first week in office was one to retitle the Gulf of Mexico on official US maps and federal communications.

The change in name will only appear for Google Maps users in the US.

Google also said its US app users would see North America’s highest mountain, Mount Denali, revert to its previous name of Mount McKinley, following a separate renaming push from Trump.

“We’ve received a few questions about naming within Google Maps,” the company said in a social media post on Monday. “We have a longstanding practice of applying name changes when they have been updated in official government sources.”

Google said the changes – for both Gulf of Mexico and Mount Denali – will be made “quickly” once the Geographic Names Information System (GNIS) is updated.

“When official names vary between countries, Maps users see their official local name,” the company said. “Everyone in the rest of the world sees both names. That applies here too.”

The name change appears to be of high priority for Trump, who mentioned it in his inaugural address on 20 January – just minutes after taking the oath of office.

“America will reclaim its rightful place as the greatest, most powerful, most respected nation on Earth, inspiring the awe and admiration of the entire world,” Trump said at the time. “A short time from now, we are going to be changing the name of the Gulf of Mexico to the Gulf of America.”

Meanwhile, Mexico’s president, Claudia Sheinbaum poked fun at Trump’s renaming decision and even initially sarcastically suggested renaming North America as “América Mexicana” or “Mexican America”.

“He says that he will call it the Gulf of America on its continental shelf,” Sheinbaum said last week. “For us it is still the Gulf of Mexico, and for the entire world it is still the Gulf of Mexico.”

0:50Watch: Mexico’s president rebukes Trump’s vow to rename Gulf of Mexico

Further north in Alaska, Trump said he will restore Mount Denali to its previous name of Mount McKinley to honour former President William McKinley who he said made the US “very rich” through tariffs and talent.

The mountain had been formally titled Denali in 2015, under President Barack Obama’s administration as a gesture to Native Americans in Alaska.

Indigenous organizations and Alaska Senator Lisa Murkowski, a Republican, have come out against Trump’s decision.

“Our nation’s tallest mountain, which has been called Denali for thousands of years, must continue to be known by the rightful name bestowed by Alaska’s Koyukon Athabascans, who have stewarded the land since time immemorial,” she said.

Asked if he would rule out using military or economic force in order to take over the autonomous Danish territory or the Canal, he responded: “No, I can’t assure you on either of those two.

“But I can say this, we need them for economic security,” he told reporters during a wide-ranging news conference at his Mar-a-Lago estate in Florida.

Both Denmark and Panama have rejected any suggestion that they would give up territory.

Trump also vowed to use “economic force” when asked if he would attempt to annex Canada and called their shared border an “artificially drawn line”.

The boundary is the world’s longest between two countries and it was established in treaties dating back to the founding of the US in the late 1700s.

The president-elect said the US spends billions of dollars protecting Canada, and he criticised imports of Canadian cars, lumber and dairy products.

“They should be a state,” he told reporters.

But outgoing Canadian Prime Minister Justin Trudeau said there isn’t “a snowball’s chance in hell” of the two countries merging.

The news conference was initially billed as an economic development announcement to unveil Dubai developer Damac Properties’ $20bn investment to build data centres in the US.

But the president-elect went on to criticise environmental regulations, the US election system, the various legal cases against him, and President Joe Biden.

Among a variety of other things, he suggested renaming the Gulf of Mexico the “Gulf of America” and restated his opposition to wind power, saying wind turbines are “driving the whales crazy”.

His remarks came as his son, Donald Trump Jr, was visiting Greenland.

Before arriving in the capital Nuuk, Trump Jr said he was going on a “personal day trip” to talk to people, and had no meetings planned with government officials.

When asked about Trump Jr’s visit to Greenland, Danish Prime Minister Mette Frederiksen told Danish TV that “Greenland belongs to the Greenlanders” and that only the local population could determine their future.

She agreed that “Greenland is not for sale”, but stressed Denmark needed close co-operation with the US, a Nato ally.

Greenland lies on the shortest route from North America to Europe and is home to a large American space facility. It also has some of the largest deposits of rare earth minerals, which are crucial in the manufacture of batteries and high-tech devices.

Trump suggested the island is crucial to military efforts to track Chinese and Russian ships, which he said are “all over the place”.

“I’m talking about protecting the free world,” he told reporters.

Since winning re-election Trump has repeatedly returned to the idea of US territorial expansion – including taking back the Panama Canal.

During the news conference, Trump said the canal “is vital to our country” and claimed “it’s being operated by China”.

He previously accused Panama of overcharging US ships to use the waterway, which connects the Atlantic and Pacific oceans.

Panamanian President José Raúl Mulino has rejected Trump’s claims and said there is “absolutely no Chinese interference” in the canal.

A Hong Kong-based company, CK Hutchison Holdings, manages two ports at the canal’s entrances.

The canal was built in the early 1900s and the US maintained control over the canal zone until 1977, when treaties negotiated under President Jimmy Carter gradually ceded the land back to Panama.

“Giving the Panama Canal to Panama was a very big mistake,” Trump said. “Look, [Carter] was a good man… But that was a big mistake.

“It’s unclear how serious the president-elect is about adding to the territory of the US, particularly when it comes to Canada, a country of 41 million people and the second-largest nation by area in the world.

During the news conference, Trump also repeated a number of falsehoods and odd conspiracy theories, including suggesting that Hezbollah, the Islamist militant group, was involved in the US Capitol riot of 2021.