President Trump appeared to suggest on Sunday that New Mexico should be renamed “New America,” prompting swift backlash from the state’s Democratic leaders over his latest effort to put an American stamp on geographic names.

On social media, Mr. Trump shared an image of a map of New Mexico with the word “Mexico” crossed out with a red X and replaced with “America.” The White House re-shared the image shortly after on its official X account.

The president posted again several hours later, with what appeared to be an A.I. video of him placing a label with the word “America” over “Mexico” on a “Welcome to New Mexico” road sign.

Mr. Trump has no legal authority to rename any state, which in New Mexico’s case is written into its Constitution. Any proposed change would have to originate within the state, a prospect that New Mexico’s Democratic leaders rejected forcefully on Sunday.

“New Mexico’s name isn’t up for debate — it’s been ours since before the United States existed,” Gov. Michelle Lujan Grisham wrote on social media.

Representative Melanie Stansbury was more blunt: “Oh hell no!” she wrote on X, adding, “Our state’s name carries generations of history, culture, and family — and it belongs to us.”

The origins of New Mexico’s name predate the founding of the United States. In the 16th century, Mexicans called the territory north and west of the Rio Grande “New Mexico,” according to the Bureau of Indian Affairs.

The state’s name is also written into Article 1 of its Constitution: “The name of this state is New Mexico, and its boundaries are as follows,” the first page reads.

New Mexico’s two Democratic senators also pushed back on Mr. Trump’s post. “Three things I will always call by their actual names …” Senator Martin Heinrich wrote on X, before listing the Gulf of Mexico, Lake Ontario and New Mexico.

On the day Mr. Trump returned to office last year, he signed an executive order directing the federal government to rename the Gulf of Mexico to the Gulf of America. The name “has a beautiful ring,” the president said that month.

Last month, amid an escalating trade fight with Canada, Mr. Trump signed another executive order to change the name of Lake Ontario to Lake America. Days before signing the order, the president had posted an image of a map with Lake Ontario crossed out with a red X, as well as an A.I. video of himself replacing a Lake Ontario sign with “Lake America.”

Those executive orders, however, dealt with geographic features, not the legal name of a state. The changes were carried out through the federal Geographic Names Information System and apply only to federal usage, including on maps and documents, but they do not bind state and local governments or private mapmakers.

Rhode Island offers a recent example of how different the process for changing the legal name of a state can be. In June 2020, amid nationwide protests over systemic racism after the killing of George Floyd, the state moved to drop “and Providence Plantations” from its formal name. Gov. Gina Raimondo, a Democrat, ordered the phrase be removed from executive-branch documents and websites while acknowledging that changing the state’s official name required voters to amend the Rhode Island Constitution. Voters approved the change that fall.