The comments he referred to related to a Senate Foreign Relations hearing on Wednesday on the confirmation of Riley Barnes, Trump's pick for assistant secretary of state for democracy, human rights and labor.
In opening remarks, Barnes quoted Secretary of State Marco Rubio as saying, "We are a nation founded on a powerful principle ... that all men are created equal, because our rights come from God our creator — not from our laws, not from our governments."
Virginia Senator Tim Kaine, who was the Democratic pick for Vice President in the 2016 Presidential campaign in which Trump was triumphant, called the assertion "extremely troubling" while adding that he believes in the idea of natural rights.
"The notion that rights don't come from laws and don't come from the government but come from the creator, that's what the Iranian government believes. it's a theocratic regime that bases its rule on Shia law and targets Sunnis, Baha'is, Jews, Christians and other religious minorities," Kaine said.
"They do it because they believe that they understand what natural rights are from their creator."
The July 4, 1776, document considered foundational to American democratic principles declares, "we hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights.
"To secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed," it adds.
Trump, who counts evangelical Christians as a key constituency, lambasted the comments in a speech at Washington DC's Bible Museum to laughter and applause from supporters.
"Isn't it terrible how he would say something like that, that this is advocated really by a totalitarian regime? This is what they say.
"But as everyone in this room understands, it's the tyrants who are denying our rights and the rights that come from God," he added, saying Kaine ought to be "ashamed of himself."