No. This is an urban myth. What is true is that Santa Claus, to give him his American name, was depicted in Coca-Cola adverts in the 1930s wearing his regulation red and white outfit, which is of course the colour of the Coke trademark. However, Santa Claus had been wearing those colours for decades before then. Illustrations of a jolly, bearded, chubby man, clad in a red suit trimmed with white fur and with a sackful of toys, were widespread from the late 1800s.
"Rather than Coca-Cola, Santa’s current image may actually be down to Charles Dickens."
There is a division between Santa Claus and Father Christmas. Santa Claus is the American version of the figure known in the UK as Father Christmas, and as Sankt Nikolaus, Sinterklaas or Pere Noël in mainland Europe. There are interesting connections and separations, but Father Christmas is a much older figure than Santa, thought to have made his first appearance in festive entertainments in the 16th century in Britain, under various names including Prince Christmas and Captain Christmas. These were the first instances of Christmas taking on a personal image and being embodied by a person rather than as a festival, and at this point he was skinny and wore a green or brown hooded robe.
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In fact, rather than Coca-Cola, Santa’s current image may be down to Charles Dickens. When the Puritans took over England in the 1640s, they effectively banned Christmas, taking it off the statute books in Scotland and then in England and sending soldiers round churches on Christmas morning to ensure no one was celebrating. When Charles II took the throne in 1660, he put Christmas back on the calendar but it remained a very muted affair, and this only increased when the Industrial Revolution split up families and flung them all over the country.
But in early Victorian times, a group of writers led by Charles Dickens decided to revive Christmas, and his contribution was A Christmas Carol, published in 1843. He originally conceived of it as a polemic about the way the poor were being treated in Victorian Britain, but then realised that he could weave this into a festive story to win a wider audience. In the illustrations for the book, the figure of the Ghost of Christmas Present was fat and jolly, and subsequent depictions of Father Christmas took on this form. Gradually his robe changed to the scarlet we know today, either because it’s eye-catching or because of the connection with holly and robins, or even because postmen around that time wore red outfits.