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Like lots of things to do with myth there are lots of theories, few of which agree with each other.
In the 19th century, English Christmas cards showed Father Christmas in blue, green, red or brown outfits. Although red was the most common. This is possibly a reflection of Father Christmas' duel pagan and Christian origins - the Green Man and St Nicholas.
In the US up until the 1860s the color of Santa's suit also were open to individual interpretation. Then in 1863, Thomas Nast, a German immigrant illustrate Clement C Moore's poem of the 1820s 'A Visit of St Nick' (Twas the Night before Christmas) [There is some doubt over Moore's authorship but that is another story.] Nast drew Santa in red and endowed him with human characteristics. Nast's illustrations of Santa thus dress appeared annually in Harpers Weekly for over 20 years.
However, the worldwide acceptance of Santa in red does seem to be due to Coca Cola. In 1931 Haddon Sundblorn drew a red dressed Santa for a billboard and this image (with some alterations) has been used by Coca Cola every Christmas since.
In short, Yes and not really.