Mail Online, which published her claim, also paid Â£150,000 in libel damages to the Mahmood family.
Hopkins wrongly said the family had links to al-Qaeda in two articles published in December 2015.
Her apology has been shared more than 3,500 times on Twitter, including by the Mahmood family's MP, Stella Creasy.
But the family told the BBC's Asian Network they wanted a personal apology by phone from Hopkins herself, adding: "She owes that to us."
The family of 11 had planned to holiday to Disneyland on 15 December 2015 but were stopped by US authorities at Gatwick Airport.
Hopkins's article from 23 December said "you can't blame America for not letting this lot travel to Disneyland - I wouldn't either".
Mail Online has now removed the story from its website, which claimed Mohammed Tariq Mahmood and his brother, Mohammed Zahid Mahmood, were extremists and published an apology.
At the time, Mohammad Tariq Mahmood said he was given no reason why US officials had refused to allow the family on board.
But Hopkins, who gained fame as a candidate in The Apprentice BBC TV series in 2007, claimed that his reason for visiting the US to go to Disneyland was a lie.
Stella Creasy MP, who represents the family's Walthamstow constituency, accused Hopkins of "tucking away" her apology, which she published on Twitter at 02:00 GMT.
"Do feel this late night tucked away one should be given more prominence," she said on Twitter.
Ms Creasy had called on the former Prime Minister, David Cameron, to challenge the US for stopping the Mahmoods from boarding their flight to Los Angeles.
Tariq Mahmood told the BBC it was "a great relief" to have the record set straight, but said he feared there would "always be feelings towards us" from "those who hate Muslims".
"Once you drill a hole in the wall, you can fill it but the mark will always be there," he added.
Hopkins, who joined the Mail Online in November last year, is known for airing controversial views, having compared migrants to "cockroaches" while a writer at the Sun.
Before she wrote about the Mahmoods, president-elect Donald Trump had already praised Hopkins for "powerful writing on the UK's Muslim problems". Hopkins replied on Twitter, saying "you have support in the UK".
In a second article published on 29 December, which included pictures of the Mahmood family home, Hopkins added that Hamza Mahmood, Mohammed's son, was publishing "extremist material" on a Facebook page.
Hamza Mahmood said he did not create the Facebook page, which was linked to him because of an error with his email address.
"We are happy to make clear that there is no suggestion that either Hamza nor Taeeba or Hafsa Mahmood (Hamza's mother and sister) have any links to extremism," Mail Online said in a statement issued on Monday.
It added: "We and Katie Hopkins apologise to the Mahmood family for the distress and embarrassment caused and have agreed to pay them substantial damages and their legal costs."
Tariq and Zahid Mahmood said: "Even to this day the US authorities have not explained the reason why we were not permitted to travel. We assume it was an error or even a case of mistaken identity.
"However, matters are not helped when such sensationalist and, frankly, Islamophobic articles such as this are published, and which caused us all a great deal of distress and anxiety.
"We are very pleased that the record has been set straight."
Columnist Katie Hopkins has apologised to a Muslim family she accused of being extremists after they were refused entry to the US for a Disneyland trip.