Bing Crosby isn't a likely person to target as perpetrating a war on Christmas. In 1943, the film Holiday Inn won an Academy Award for best original song for "White Christmas." But that same movie opened with Bing singing Happy Holiday. There's a scene in which a small black child dressed up as Father Time joins in the song. If it had been made today, Fox News would certainly be taking the movie to task.
Wikipedia has several articles that speculate that the phrase "happy holidays" as a greeting was popularized by Holiday Inn. (here, and here). That would be ironic if true, that the same movie that gave us "White Christmas" also popularized a now polarizing phrase. I can't find any proof that the film is responsible for the popularity of the phrase. But, while I admit the written corpus that Google NGrams uses might be quite different than verbal usage, plotting "Happy Holidays" and "Merry Christmas" should be revealing. It does show an increase in popularity of the term "happy holidays" after 1944. It also shows a corresponding drop in the use of "merry christmas" which continues on until the 1970s.
So the words "happy" and "holidays" start appearing in Google's corpus around 1806. From what I can tell from a cursory investigation, throughout most of the 1800's, the meaning is usually like having a pleasant vacation. But over time the term begins to evolve so that it's more directly associated with Christmas. A case sensitive search for "Happy Holidays" makes things a bit clearer - around the 1860s the phrase appears to be clearly about the Christmas and New Year season.
A blogger, Jeremy Aldrich, collected some early ads that use the phrase Happy Holidays. These date back to the 1860's. They aren't quite the same as modern usage - "happy holidays are coming" and "hail happy holidays!" aren't the same as the greeting "Happy Holidays", but are recognizable.
So "happy holidays" is rather old. From my memory, the offense at the phrase is quite new - less than a decade old in fact - and is a sign that Fox News is winning their manufactured war.