No, Christmas does not fall on a Wednesday between 2004 and 2014. However, it does fall on a Wednesday during2014, so it is true that Christmas falls on every day of the week at least once during the period of 2004 through 2014 (inclusive).
Christmas falls on every day of the week between 2004 and 2014.
Before Christmas - 2014 was released on: USA: 2014 (limited)
Merry Friggin' Christmas - 2014 was released on: USA: 2014
Christmas in Palm Springs - 2014 was released on: USA: 2014
My Christmas Home - 2014 was released on: USA: 2014 (TV premiere)
Christmas on the Gulf - 2014 was released on: USA: 1 January 2014
In the year 2014, Christmas Day will be on a Thursday.
Christmas Surprise - 2014 was released on: USA: 6 January 2014 (limited)
The Christmas Thief - 2014 was released on: USA: 2 February 2014 (internet)
Christmas with the Karountzoses - 2014 was released on: USA: 1 June 2014 (video premiere)
Christmas The Aftermath - 2014 was released on: UK: 1 January 2014 (internet) USA: 1 January 2014 (internet)
Christmas Tree Farm - 2014 TV was released on: USA: November 2014
A Very Grimm Christmas - 2014 was released on: USA: 1 January 2014 (limited)