Mr Cook said he did not agree there was a "moral equivalence" between white supremacists and "those who oppose them".
Mr Trump has disbanded two business councils after top bosses resigned.
Mr Cook said Apple will also make donations to human rights charities.
In an email to staff obtained by Buzzfeed News Mr Cook said: "I disagree with the president and others who believe that there is a moral equivalence between white supremacists and Nazis, and those who oppose them by standing up for human rights.
"Equating the two runs counter to our ideals as Americans."
He added that "in the wake of the tragic and repulsive events in Charlottesville, we are stepping up to help organisations who work to rid our country of hate".
Apple will donate $1m to both the Southern Poverty Law Center and the Anti-Defamation League. It will also match two-for-one any staff donations to these and several other human rights groups until 30 September, Mr Cook said.
On Wednesday, Mr Trump has said he was scrapping two business councils after more bosses quit over his handling of the violent clashes in Virginia.
Trump ends business groups as CEOs quit
A White House meltdown in the making
Business mutiny leaves Trump isolated
Business leaders left the White House manufacturing council after the backlash against how he reacted to the far-right rally last weekend.
The clashes culminated in a woman's death and nearly 20 wounded when a car ploughed into a crowd of anti-fascist protesters.
Mr Trump's reaction has sparked outrage and generated global headlines.
His announcement on Twitter came as the heads of 3M, Campbell Soup, Johnson & Johnson and United Technologies announced their resignations on Wednesday.
Campbell Soup Co President Denise MorrisonImage copyrightGETTY IMAGES
Image caption
Denise Morrison of Campbell Soup was the eighth CEO to resign from a White House business panel
Mr Trump said: "Rather than putting pressure on the businesspeople of the Manufacturing Council & Strategy & Policy Forum, I am ending both."
Before Mr Trump's announcement, the Strategy and Policy Forum announced it was a joint decision to disband the council.
Businesses have been under pressure to distance themselves from Mr Trump over his handling of the clashes in Charlottesville, Virginia.
Why did they leave?
On Monday, Mr Trump belatedly condemned the white supremacist and neo-Nazi groups that rallied in a small Virginia town on Sunday.
But in a rancorous news conference on Tuesday he backtracked and again blamed left-wing counter-protesters for the violence too.
JPMorgan chief executive Jamie Dimon, a member of the Strategy and Policy Forum, released a separate statement on Wednesday saying he strongly disagreed with Mr Trump's recent statements, adding that "fanning divisiveness is not the answer".
"Constructive economic and regulatory policies are not enough and will not matter if we do not address the divisions in our country. It is a leader's role, in business or government, to bring people together, not tear them apart," he said.
Denise Morrison of Campbell Soup said she could not continue to participate in the advisory panel after Mr Trump's comments. Activists had called on Campbell Soup, among other firms, to take action.
