Speaking earlier in Miami, Florida, Mr Trump said he was re-imposing 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".
Barack Obama attempted to thaw relations by loosening trade and tourism bans.
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.