In video footage posted online on Saturday, the woman is heard asking a flight attendant if the man sitting next to her could be moved to another seat.
She tells the man next to her: "You pretend you have the moral high ground but you put that man's finger on the nuclear button."
An employee on the plane is heard telling the woman that she will have to leave the aircraft, but she pleads, saying she will be quiet and that she is traveling with her husband whose mother has died.
However, her pleas are to no avail, with a separate clip showing her being escorted off the plane by a police officer to the excitement of the other passengers.
The woman was later escorted off the plane by a police officer
That man [Mr Trump] doesn't believe in climate change. Do you believe in gravity? Did you know gravity's just a theory?
Woman on plane
Scott Koteskey, who posted the video to Facebook the day after Mr Trump's inauguration, called the incident his "craziest experience ever on an airplane".
He said he was the man in the video and that the incident unfolded after he said he had come to celebrate.
In the clip, the flight attendant asks: "Is there going to be a problem?"
But the woman tells her that she would like the man to change seats.
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President Donald Trump waves as he walks with first lady Melania Trump and son Barron during the inauguration parade.
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"Well, you don't have that right," the flight attendant replies, before saying she will get someone else to deal with the issue.
During the exchange with the man, the enraged woman says: "'That man [Mr Trump] doesn't believe in climate change. Do you believe in gravity? Did you know gravity's just a theory?"
Another member of staff arrives and tells the woman she will have to leave the aircraft.
Pointing to her husband, the woman says: 'I'm going home now. My mother-in-law – his mother died."
The second video shared by Mr Koteskey shows the woman being taken away and off the plane, with other passengers audibly pleased at her departure.