The incident took place at a local Apple store when the woman named Shree Chauhan saw Spicer and started questioning him as she filmed the whole sequence.
The video was posted by her on Twitter on Sunday and has gone viral.
Reacting to the incident, Spicer, at his daily news conference yesterday, said that the US is a free country and people have the right to act however they want.
"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.
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"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.
As she questioned Spicer multiple times, he replied saying "such a great country that allows you to be here", which the Indian-American described as racist remarks.
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.
The incident has drawn criticism from a top
Indian-American donor of Trump's presidential campaign, who called the woman's behaviour as disgusting.