"In Dow there is no room for hatred, racism, or bigotry," Liveris said. "Dow will continue to work to strengthen the social and economic fabric of the communities where it operates — including supporting policies that help create employment opportunities in manufacturing and rebuild the American workforce."

Immelt recently stepped down as chief executive of GE, but he is expected to remain the company's chairman until the end of the year.

" GE has no tolerance for hate, bigotry or racism, and we strongly condemn the violent extremism in Charlottesville over the weekend," the company said in a statement. "With more than 100,000 employees in the United States, it is important for GE to participate in the discussion on how to drive growth and productivity in the U.S., therefore, Jeff Immelt will remain on the Presidential Committee on American Manufacturing while he is the Chairman of GE."

CNBC.com reached out to every executive listed on the White House's press release from late January, when the members of the council were announced, and only seven had issued comments on either the recent violence in Charlottesville, Virginia , or Frazier's resignation by Monday afternoon. Many of those comments came hours after Frazier issued his statement at 8 a.m. ET.

While the president responded quickly via Twitter to the decision by Frazier, the move by the Merck CEO was met by silence from many other executives on the council.

The chief executives of Merck , Under Armour and Intel all resigned in quick order from President Donald Trump 's American Manufacturing Council on Monday, with Merck's Kenneth Frazier citing his "responsibility to take a stand against intolerance and extremism."

Dow Chemical will remain on the council as will Whirlpool.

The appliance manufacturer said it wants to continue to work on the Manufacturing Jobs Initiative, and "to provide input and advice on ways to create jobs and strengthen U.S. manufacturing competitiveness." But added that it supports an "open and inclusive culture that respects people of all races and backgrounds."

John Ferriola, CEO of Nucor, said the company condemns the violence over the weekend and he will continue to serve on the council, saying in a statement, "We believe a strong manufacturing sector is the backbone of a strong economy."

International Paper spokesperson Tom Ryan echoed the response from Ferriola condemning the violence over the weekend and adding the company "fosters an inclusive workforce."

Campbell Soup CEO Denise Morrison will remain on the council to "have a voice and provide input" on matters concerning the company, Campbell Soup said in a statement.

Dell also indicated no change in engagement with the administration for its CEO, Michael Dell.

Several executives have already left the president's advisory groups for other reasons, including former Uber CEO Travis Kalanick, who resigned in February over the administration's immigration policies, and Tesla CEO Elon Musk and Disney's Bob Iger, who both left in June after Trump said he would withdraw from the Paris climate accord.

In addition, several executives were no longer part of the council as they were no longer CEOs. They include: Mark Fields of Ford, Klaus Kleinfeld of Arconic, and Mario Longhi of U.S. Steel.

Times like these can be "a real test of leadership character" for executives, said management expert Jeffrey Sonnenfeld on CNBC's "Squawk Alley" on Monday. "There's no blueprint here and we can see which CEOs are passing the exam on this one and which ones are failing," he said.

Several other CEOs who weren't part of the manufacturing council did comment on Frazier's resignation.

Meg Whitman, CEO of Hewlett Packard Enterprise CEO, said she supported Frazier's decision.

"I'm thankful we have business leaders such as Ken to remind America of its better angels," Whitman continued.

@MegWhitman: "I support Ken Frazier's decision. I'm thankful we have business leaders such as Ken to remind America of its better angels."

Paul Polman, CEO of Unilever, also supported the decision, saying it showed "strong leadership."

@PaulPolman: Thanks @Merck Ken Frazier for strong leadership to stand up for the moral values that made this country what it is

At PepsiCo, CEO Indra Nooyi said she was "heartbroken" by the violence from the weekend, but she did not make any reference to Frazier's resignation.

@IndraNooyi: Heartbroken by the violence in #Charlottesville. Hate and intolerance are a betrayal of what we stand for as Americans.

Meanwhile, former Merck CEO Roy Vagelos said in a statement he supports his protege's decision.

"He is always smart, always ethical and repeatedly makes the right decisions," Vagelos said in the statement. "I applaud his decision to step down from the Council this morning. Ken is driven by a strong sense of morality in everything he does and he continues to make me proud," the statement read.

Frazier "is truly the American dream," said Sonnenfeld, noting his rise from poverty to C-suite executive and his leadership on combating abuses in drug pricing. "How could you pick on a worse target to fire back on than the president picking on Ken Frazier?"

Sonnenfeld was referring to the response Trump gave Monday morning, that said Frazier's resignation would give him more time to focus on lowering "ripoff" drug prices.

@realDonaldTrump: Now that Ken Frazier of Merck Pharma has resigned from President's Manufacturing Council,he will have more time to LOWER RIPOFF DRUG PRICES!

Despite the harshness of the comments, which recalled Trump statements earlier this year that he planned to work to lower drug prices, Merck shares were mostly flat to higher in Monday trading and closed up modestly at $62.67.

That's a much different response from the early days of the administration, when the president's negative tweets about companies usually sent shares tumbling and inspired pieces like FTI's "If POTUS Tweets @ Your Company: Six Steps."

The list included advice like consider laying low, addressing misinformation and being proactive on social media. FTI also advised engaging with Trump offline, in person, to allow for a more substantive response.

Reflecting on the discussion, former McDonald's CEO, Ed Rensi, told CNBC's "Squawk Alley" on Monday that Trump needs to watch out.

"He'll become irrelevant if he doesn't' change his behavior," Rensi said.

The president was roundly criticized Saturday for suggesting "many sides" were to blame for the deadly violence at the rally in Charlottesville. Members of both parties lambasted Trump, saying his comments failed to single out white nationalists.

On Monday, speaking at the White House, Trump was more specific, saying, "Racism is evil and those who cause violence in its name are criminals and thugs, including the KKK, neo-Nazis, white supremacists and other hate groups that are repugnant to everything we hold dear as Americans."

While Sonnenfeld said many executives had contacted him privately bemoaning the situation, he urged them take their comments public.

"Is...[former Starbucks CEO] Howard Schultz sipping on his cappuccino and thinking of more clever things to write on coffee cups? Why not get out there and say something?" Sonnenfeld said.