CAIR said Rose Hamid's eviction from the rally in South Carolina sent a "chilling message to American Muslims".
Ms Hamid was removed after making a silent protest on Friday.
Mr Trump's proposed ban on Muslims entering the US has been widely condemned.
During the rally in the town of Rock Hill, Ms Hamid stood up wearing a T-shirt bearing the words: "Salam, I come in peace" - "Salam" is the Arabic word for peace.
When she stood up, the crowd around her began chanting "Trump! Trump!" as they had reportedly been instructed to do in the event of any interruption, reports the BBC's Anthony Zurcher from Rock Hill.
Ms Hamid, a 56-year-old flight attendant, was then told by a security officer that she had to leave. She was booed as she was escorted out.
CAIR said that as well as apologising to Ms Hamid, Mr Trump should "make a clear statement that American Muslims are welcome as fellow citizens and as participants in the nation's political process".
Mr Trump has yet to respond.
At a Republican presidential forum in nearby Columbia,  on Saturday morning, Ohio Governor John Kasich condemned the way the Trump crowd handled the protest.
"We don't need to be shouting and booing and scaring somebody," Mr Kasich, who is competing against Mr Trump for the Republican presidential nomination, told the BBC. "This is not some high-school basketball game. We are all our brother's keeper."
He added that the next US president must be willing to work with moderate Muslim nations.
"That may not win me the Republican primary, but there's more to life than that," he added.

Under new anti-terror measures, a 2.6-metre barrier could soon block public access to the much-loved grassy space.
The mass tumble took place on Saturday morning, after news of the event spread on Facebook.
Lester Yao, who organised it, said he felt the architect had wanted the public to enjoy the grounds freely.
While he had originally planned the roll as a gathering of family and friends, he was delighted when almost 3,000 people expressed interest in the unique protest via Facebook.
Cricket fans treated to spectacular Canberra sunset
Pro-asylum protest halts Australia's parliament
Syrian refugee gets top results in Oz
Explaining the event, he wrote: "It's a story I hear often from people that live in Canberra or visit Canberra and the Parliament House. They walk up that beautiful green lawn and see the amazing view of Canberra, then they fall to their knees and roll down the hill.
"This simple fun action embodies a very simple yet powerful symbol of democracy. That the citizens can walk up and over their elected government.
"At least that's what I believe the architect behind the design of the Parliament House intended."
Mr Yao, 34, lives in Canberra but is originally from the Philippines. The Canberra Times reported that he recently took a visiting cousin for a joyful tumble down the Parliament House hill - which got the ball rolling on his protest.
No firm timetable has been given for the controversial fence, which is part of a A$60m (Â£35m; $44m) security blitz at the seat of Australia's government, but work could begin by summer 2017.
Prime Minister Malcolm Turnbull has defended the costly plans, telling The Huffington Post Australia that "parliament is the people's house," but that "getting that balance right" on security is "critically important".
MPs recently voted in favour of the fence - but not all of them support it.
MP Andrew Broad of the National Party of Australia told News.com.au children should be able to roll down the iconic grass.
"Let's not let terrorists destroy our way of life," he urged.
Mr Broad said that while he has not indulged in a hill roll since he was a child, Australia's leaders should consider taking part.
"I reckon at the end of the year that'd be a good look; all the pollies [politicians] finish the parliament with a good roll down the grass and tell people that parliament ultimately is about a bit of fun," he said.
The Council on American-Islamic Relations (CAIR) has urged Republican presidential hopeful Donald Trump to apologise to a Muslim protester ejected from one of his rallies.
[NEXT_CONCEPT]
Hundreds of Australians have gathered to roll on the lawns of Parliament House in Canberra, in protest against plans to build a security fence there.