Shree Chauhan, 33, challenged Mr Spicer as he was shopping, asking: "How does it feel to work for a fascist?" - referring to President Donald Trump.
Mr Spicer said: "It's such a great country that allows you to be here."
Ms Chauhan videoed the encounter and posted it online, drawing 300,000 views since Saturday.
She has been attacked online since posting the video, with critics branding her "vile" and accusing her of harassment.