Chauhan also asked Spicer about Russia and accused President Donald Trump of committing treason.
In the video, Chauhan put out on various social media outlets that have gone viral, Spicer is heard saying that the US is "such a great country that allows you to be here".
Chauhan describes his remarks as racist.
"That is racism and it is an implied threat. Think about the sheer audacity of Mr Spicer to say that to my face with a smile, knowing that he is being recorded on video and the position of power he holds in our government," she said.
The White House Secretary told reporters at his daily news conference that the United States is a free country and people have right to act how they want.
As per Chauhan's social media posts, she was in the store to get her iPhone fixed when she spotted Spicer.
"I realised what an enormous opportunity it was to get answers without the protections normally given to Mr Spicer. I was honestly quite nervous and wanted to come up with more cogent questions but did not have time to do so," she wrote in a post on Medium.Com.
According to the video, she asked Spicer, "Have you helped with the Russia stuff? Have you committed treason too, just like the president? What can you tell me about Russia....And How do you feel about destroying our country, Sean?"
She said she has lived in Washington DC for about a decade and during this time she has met several VVIPs at public places including pharmacy and grocery stores but did not interact with them.
"However, given what Mr Spicer and his boss (Donald Trump) are doing to this country, I do not believe they are entitled to these norms and customs. Donald Trump and his Klan are openly trampling on the rule of law, our Constitution and our democracy," she said.
At the news conference, Spicer said if people have questions, they should ask him.
"Ask it. I interact with individuals all day long. Ninety-nine percent of them are pleasant, even with people who may not agree with our philosophy or programs or whatever," he said.
"But it's a free country and the beauty of it is that people can act how they want no matter how that's interpreted, and as long as they step on the right side of the First Amendment, we're good," he added.