The video, uploaded to Facebook and titled "Starbucks removed Christmas from their cups because they hate Jesus", has been viewed millions of times.
Joshua Feuerstein, a Christian evangelist from Phoenix, Arizona, is angry that the company has chosen a minimalist design to its cups this year, in contrast to its usual more festive choice.
More than 15 million people have seen the rant in which Mr Feuerstein urges Christians to give their name to baristas as "Merry Christmas" so the staff are forced to write it on the cup.
"I think in the age of political correctness we've become so open-minded our brains have literally fallen out of our head," he said.
"Do you realise that Starbucks wanted to take Christ and Christmas off of their brand new cups? That's why they're just plain red.
"In fact, do you realise Starbucks isn't allowed to say Merry Christmas to customers?
"So I decided instead of simply boycotting, well why don't we start a movement.
"So when I went in and asked for my coffee and they asked for my name and told I told them my name is 'Merry Christmas'.
"So guess what Starbucks? I tricked you into putting Merry Christmas on your cup."
Elsewhere US Presidential candidate Donald Trump is calling for a boycott of Starbucks because of the cup.
"I have one of the most successful Starbucks in Trump Tower," he told a crowd at a rally.
"Maybe we should boycott Starbucks? I don't know. Seriously, I don't care. By the way, that's the end of that lease but who cares."
"If I become president, we're all going to be saying Merry Christmas again, that I can tell you. That I can tell you. Unbelievable," he added.
Starbucks has not commented on the outrage caused by the red cup but did make a statement when it was first launched.
Jeffrey Fields, vice president of design and content, said: "In the past, we have told stories with our holiday cup designs.
"This year we wanted to usher in the holidays with a purity of design that welcomes all of our stories.
"Starbucks has become a place of sanctuary during the holidays.
"We're embracing the simplicity and the quietness of it. It's a more open way to usher in the holiday."