However, Cuba said it would still cooperate with its larger neighbour.
Speaking earlier in Miami, Florida, Mr Trump said he was reimposing certain travel and trade restrictions eased by the Obama administration, condemning a "completely one-sided deal".
But he is not reversing key diplomatic and commercial ties.
"The government of Cuba denounces the new measures toughening the embargo," Cuban state TV said.
But it also reiterated "its willingness to continue the respectful dialogue and cooperation".
Mr Obama attempted to thaw relations by loosening trade and tourism bans.
President Trump said his new policy would tighten rules on travel and on sending funds to Cuba.
Announcing the changes on Friday, he cited human rights concerns, saying doing a deal with the "brutal" Castro government was "terrible" and "misguided".
Later that night, Cuban state news was also talking tough.
"Any strategy aimed at changing the political, economic and social system in Cuba - whether by pressure or imposition or through more subtle means - is destined to fail," it said.
However, President Trump is not rowing back on all parts of Obama's deal.
He will not close the US embassy in Havana, commercial flights from the US will continue, and Americans will still be able to return home with Cuban goods.
A history of the US trade embargo with Cuba
1959: Cuban revolutionary Fidel Castro leads a guerrilla army into Havana overthrowing the Batista regime.
1960: In response to Castro's communist reforms, US breaks off diplomatic relations with Cuba and imposes a trade embargo.
1962: Castro agrees to allow the Soviet Union to deploy nuclear missiles on the island bringing the US and the USSR to the brink of nuclear war.
April 2009: President Barack Obama lifts restrictions on family travel and the sending of remittances to Cuba.
July 2015: The US and Cuba reopen embassies in each other's capitals and restore full diplomatic ties.
March 2016: President Obama makes a three-day visit to Cuba and holds talks with President Raul Castro. He expresses hope the embargo will be ended, but it can only be lifted by the US Congress which is controlled by Republicans who oppose the move.
Aug 2016: US commercial flight arrives in Cuba for the first time in more than half a century.
Photo: AFP US President Donald Trump speaks in Little Havana, Florida about policy changes he is making toward Cuba.