On Tuesday, after the Supreme Court upheld lower court decisions and ruled in her favor,
Miller said she hoped for an end to the abuse that accompanied each court victory.
"In Britain we are lucky, we are fortunate, to have the ability to voice legitimate concerns and views as part of a shared society. I have therefore been shocked by the levels of personal abuse that I have received from many quarters over the last seven months for simply bringing and asking a legitimate question," Miller said outside court after the ruling, where she was accompanied by bodyguards.
Speaking to CNN later, she said that "everyone, everywhere should be standing up against that sort of abuse."
"When it comes to the abuse, I do believe that if I was a white male, yes, I'd get abuse, but the personal abuse I've got, because of my sex -- because of my gender -- and because of my race is totally, totally unacceptable," she told CNN's Paula Newton, who is hosting the "Amanpour" show today.
Lived in UK for decades
Miller is the most prominent claimant in the legal case that challenged the government's plan to bypass parliament and begin the process of pulling Britain out of the European Union.
She was born in Guyana and has lived in the UK for more than four decades, since the age of 10. She runs an investment company with her husband, Alan Miller. Much of the abuse Miller received has been racial in character.
Miller thanked those who had supported her, and called for quicker and more prominent condemnation of abuse.
"I sincerely hope that going forward, people who stand in positions of power and profile are much quicker in condemning those who cross the lines of common decency and mutual respect," she said outside the court.
"I would like to whole-heartedly thank those who've sent me the most heart-warming messages of support. They have truly helped to bolster me in this most arduous process."
Miller called Brexit the "most divisive issue of a generation." The campaign period marred by a spike in racially motivated attacks
and some minority ethnic groups complained about the way immigration was being debated.
"But this case was about the legal process, not politics," she said.
Miller told CNN that her case was aimed at ensuring parliamentary scrutiny over the Brexit process and was not an attempt to challenge the referendum results.
"This was about having a debate and a vote and act of parliament -- it is exactly what this case is about. It's about parliament being sovereign and doing the job they are elected to do," she said.
"We cannot have governments behaving as if they're dictatorships -- that's what this case is about."
Miller set up the True and Fair campaign in 2012 to promote transparency in the financial sector to avoid scandals and to provide consumer protection in the industry.
She is also a philanthropist, and she and her husband have given thousands of pounds to charities in the UK, including the Lady Thatcher infirmary at the Royal Hospital Chelsea.
Miller welcomed Tuesday's Supreme Court ruling, saying that elected members of parliament would be able to steer the government to "select the best course in the forthcoming Brexit negotiations -- negotiation that will frame our place in the world, and all our destinies to come."
"Only parliament can grant rights to the British people and only parliament can take them away. No prime minister and no government can expect to be unanswerable or unchallenged. Parliament alone is sovereign," she said.