The White House said these "common sense restrictions" would "protect Americans from dangerous foreign actors".
In a video posted to his Truth Social website, Trump said the recent alleged terror attack in Boulder, Colorado "underscored the extreme dangers" posed by foreign nationals who had not been "properly vetted".
Twelve people were injured in Colorado on Sunday when a man attacked a group gathering in support of Israeli hostages, throwing two incendiary devices and using a makeshift flamethrower.
The man accused of carrying out the attack has been identified as an Egyptian national, but Egypt is not included on the list of banned countries.
Trump has close ties with Egyptian President Abdel Fattah al-Sisi, who has in the past been described by the US president as his "favourite dictator".
Only two of the 19 countries are on the US government's State Sponsors of Terrorism list - Iran, which has a full ban, and Cuba, which faces partial travel restrictions.
But national security is given as a partial reason for the choices.
In the proclamation, Trump said many of the countries listed have a "historic failure to accept back their removable nationals" from the US, as well as having "taken advantage" of the US by exploiting its visa system.
He added that nationals from certain countries also "pose significant risks" of overstaying their visas.