Likewise, Campbell Soup Company said racist ideology must be condemned, but said its chief executive would remain on the manufacturing council because it was important to provide input on matters that affect the soup maker's industry.
At the rally, activist Heather Hayer was struck and killed and over 19 people left injured by a auto driven seemingly deliberately into counter-protesters.
The president denied that his original statement about the violence in Virginia on Saturday - saying "many" sides were to blame, rather than hate groups - was the cause of the departures.
At least four CEOs have resigned from various presidential advisory bodies in the past few days, with other corporate leaders, including the CEO of Wal-Mart, strongly rebuking the president's original statement.
"America's leaders must honour our fundamental values by clearly rejecting expressions of hatred, bigotry and group supremacy, which run counter to the American ideal that all people are created equal".
He lashed out nearly immediately on Monday following the resignation, saying on Twitter that Frazier will now "have more time to LOWER RIPOFF DRUG PRICES!"
William Galston, a senior fellow at the Washington, D.C. -based think tank Brookings Institution, said he couldn't "think of a parallel example" of any president responding as viciously as Trump to a CEO departing an advisory council.
A spokesman for Merck said the company had no further comment about Frazier's decision, his tweet or the president's response. The White House later stated that Trump was including "white supremacists, KKK Neo-Nazi and all extremist groups" in his remarks about the violence.
Frazier did not mention the violence in Charlottesville, Virginia, in his statement, but he did say that "as a matter of personal conscience, I feel a responsibility to take a stand against intolerance and extremism". "In this time of tumult in our country, Americans deserve a leader that will bring us all together and denounce those who seek to tear us apart".
The White House tried to stem the damage on Sunday.
President Trump on Tuesday ripped three top corporate chief executives who resigned from his manufacturing council in protest of his handling of the Charlottesville, Va., violence, calling them "grandstanders" and saying they were embarrassed because they made their products overseas. CEO Elon Musk resigned from the manufacturing council in June, and two other advisory groups to the president, after the US withdrawal from the Paris climate agreement.
Mr Musk has also left the manufacturing council.
And even as Republican senators and others in Mr. Trump's own party criticized him for his initial response to the violence on Sunday, other members of Mr. Trump's manufacturing council declined to follow Mr. Frazier's lead.
Former Uber Technologies Inc [UBER.UL] CEO Travis Kalanick quit the business advisory council in February amid pressure from activists and employees who opposed the administration's immigration policies.
Earlier: A prominent USA white nationalist website which promoted a Virginia rally that ended in deadly violence on Saturday has lost its internet domain host.
Free Eclipse Glasses at Wayne County Library
Those puffy clouds could block the sun at times , but should not block out the eclipse viewing for an extended period. The most recent solar eclipse was on February 26, 1979, but it only crossed above five states in the northwest.
Israeli leader criticized for response to Charlottesville
Trump's response to the events in Charlottesville has begun to attract concern in Israel, even among politicians who admire him. Marchers passed by carrying flags with swastikas and shouting the Nazi salute "Sieg Heil", the synagogue president said.