Some disagreed with the move, but others, like Sen. Rick Santorum, said that Trump did nothing wrong in failing to correct the audience member.
The first to speak out against Trump was a Democrat -- but now he's getting questioned by Republicans as well.
Democratic frontrunner Hillary Clinton tweeted about the incident, which happened during a town hall event in New Hampshire on Thursday night.
Donald Trump not denouncing false statements about POTUS & hateful rhetoric about Muslims is disturbing, & just plain wrong. Cut it out. -H— Hillary Clinton (@HillaryClinton) September 18, 2015
Sen. Bernie Sanders, who is one of the Democratic candidates in the race, jumped into the fray this morning, calling for Trump to apologize.
This nonsense has got to end.— Bernie Sanders (@BernieSanders) September 18, 2015
Let’s stop the racism. Let’s stop the xenophobia.— Bernie Sanders (@BernieSanders) September 18, 2015
Former Maryland Gov. Martin O'Malley was the third Democratic candidate to speak out against Trump, also via tweet.
Shame on you, @realDonaldTrump. “Muslim” is not a slur. You cannot scapegoat your way to the presidency. -O'M— Martin O'Malley (@MartinOMalley) September 18, 2015
New Jersey Gov. Chris Christie spoke out on the "Today Show" against Trump's decision not to correct the man.
"If somebody at one of my town hall meetings said that, I would correct them and I would say, 'The President's a Christian and he was born in this country. Those two things are self-evident,'" Christie said this morning.
"I'll tell you what I would do, and I wouldn't have permitted that if someone brought that up at a town hall meeting of mine; I would have said, 'No, listen. Before we answer, let's clear some things up for the rest of the audience.' And I think you have an obligation as a leader to do that," Christie said on Today.
South Carolina Sen. Lindsey Graham became the second Republican to rebuke Trump, when he said, during an interview on "Andrea Mitchell Reports," that Trump should apologize on live television.
When asked if he believed the Trump campaign's explanation that Trump was responding to the second part of the audience member's comment, Graham said "Oh come on, give me a break."
"You had a chance here to show who you were. This happens to all of us. It happened to John McCain. You have to push back. We are trying to be the leader of a nation here," Graham said.
The McCain moment that Graham is referring to came in the 2008 presidential campaign, when an audience member made similar claims while speaking with Sen. John McCain. In contrast to Trump's in action, at the time, McCain immediately corrected the audience member and defended Obama.
Ohio Gov. John Kasich has not commented publicly about the episode, but his spokesman Chris Schrimpf said "I think it's safe to say that the president is both American and a Christian." Kasich's team confirmed that is reflective of Kasich's thinking.
Former Sen. Rick Santorum, a fellow GOP candidate, appeared to support Trump when he said that Trump did nothing wrong in choosing not to correct the audience member.
"It's not my job... It's not Donald Trump's job to police a questioner," Santorum said this afternoon.
"I have a lot of events where people get up and say things that I don't like... look at my Twitter feed...but I am going to defend their right to say it," he said.
"Yes they have a right to say it and no we don't have an obligation to correct it," he said.
On the topic of the President's religion, Santorum's response was that Obama "says he is a Christian."
Sen. Rand Paul, Sen. Ted Cruz and Dr. Ben Carson have all commented on the incident, but did not directly criticize or praise Trump’s handling of the situation.
This is far from the first controversial moment in Trump's campaign, though many of the other incidents stemmed from comments that Trump made directly. In this case, he appeared to let the audience member slide with his claims about the president's religion and nationality.
Trump has a history of taking issue with Obama's background. In spite of the fact that the President was born in Hawaii, Trump launched a campaign in 2011 calling for the president to release his birth certificate, which President Obama did in April 2011.