Trump, meanwhile, will hold another Independence Day celebration in Washington DC on Saturday.
His political supporters say there will be no social distancing at the event, but masks will be available.
Trump is expected to speak at the event, and will criticise those attempting to "tear down" United States history with "cancel culture".
Randy Seiler, who heads the state Democratic party, told CNN the planned festivities are problematic: That they are offensive to Native Americans who consider the land sacred, that the fireworks pose a fire risk in the extremely dry area, and that the virus danger is real.
In his speech, the president took aim at "the anarchists, the agitators, the looters" who, he claimed, won't be allowed to "tear down our statues, erase our history, indoctrinate our children", referring to recent protests sparked by the death of George Floyd.
Speaking to Fox News this week, Ms Noem said free face masks would be available at the outdoor event for people who chose to wear them, but "we won't be social distancing".
"We can speak, write, worship, defend, even protest as we see fit because of these men and they're ideals", Noem says.
Why is the location controversial?Faces of Modern America's "Founding Fathers" George Washington and Thomas Jefferson are chiseled into the granite mountainside along with that of Abraham Lincoln and Theodore Roosevelt.
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The president's events at Mt. Rushmore comes amidst a renewed national debate about racism and the country's early ties to slavery.
In many major American cities the traditional fireworks, concerts and marches have been cancelled, fireworks took place only in NY, Washington D.C. and mount Rushmore in South Dakota, where the rocks are carved sculptural portraits of four American presidents.
A standoff ensued, with police using pepper spray on several protesters but taking no further action for several hours.
Young, who became a U.S. citizen this year, said he began his path to United States citizenship because he wanted to vote against Trump in the 2020 election.
The speech and fireworks at Mount Rushmore came against the backdrop of a pandemic that has killed over 125,000 Americans. (The Sandcastle is said to have recently become the most-expensive summer rental in South Fork history, when Mr. Farrell rented it to a tenant from March until Labor Day for a reported sum of almost $2 million.) Ms. Guilfoyle and the younger Mr. Trump own a house not far away in Bridgehampton. Researchers believe someone with Covid-19 becomes contagious two or three days before the onset of symptoms; however, those without symptoms can also transmit the disease.
On Friday, the USA recorded its largest single-day total of new coronavirus cases since the start of the pandemic. He said the event was a tribute to the "tremendous courage and spirit" of front-line workers and the public in the pandemic.
During the July 3 event held at the base of Mount Rushmore in South Dakota, Neil Young tracks including "Like A Hurricane" and "Rockin' In The Free World" were played both before and during the event itself.
"The president is putting our tribal members at risk to stage a photo op at one of our most sacred sites", Harold Frazier, chairman of the Cheyenne River Sioux Tribe, told AP.