"We call on the American government and their judicial authorities to put an end to the detention of these individuals," foreign ministry spokeswoman Marzieh Afkham said.
"None of the human rights institutes, including the United Nations rapporteurs, have followed up the situation of Iranian prisoners," she told reporters at her weekly press briefing.
On Tuesday, Hossein Ghashghavi, a deputy foreign minister, said the 19 Iranians "are innocent and we hope that conditions are prepared for their release".
He rejected any link with the case of Washington Post reporter Jason Rezaian, a dual American-Iranian citizen.
"There is no question at all of an exchange of prisoners as each of these cases has its own special circumstances," he said.
Rezaian, the Post's Tehran correspondent, has been in custody since he was detained at his home in the Iranian capital on July 22, 2014.
He has been tried behind closed doors for "espionage, collaboration with hostile governments, gathering classified information and disseminating propaganda against the Islamic republic."
But no verdict has been announced, with Iranian officials saying the proceedings are ongoing.
The Post has called the trial a "sick brew of farce and tragedy" and said it "has been anything but transparent and just".
Earlier this month the fourth and final hearing in the trial -- the first since Iran struck a nuclear deal with world powers in Vienna on July 14 -- centred on the reporter's rebuttal of the charges.