The band's 1987 Christmas hit featuring Kirsty MacColl has frequently come under fire for its use of homophobic slur 'f*****', which will now not feature in the alternate version on BBC Radio 1.
The original song is still set to be played on Radio 2 with 6 Music presenters having a choice of the two.
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"Here we go again," Fox wrote on Twitter. "The cultural commissars at the @bbc are telling you what is and isn’t appropriate for your ignorant little ears.
"Wouldn’t it be nice if we sent the (proper) version to the top of the charts? #DefundTheBBC. RT."
However, a message from The Pogue's Twitter account hit back at the actor's comments.
"F*** off you little herrenvolk s****," the reply read.
"Herrenvolk" is a concept from Nazi ideology referring to the "master race" they consider to be innately superior.
It comes as Fox has found himself at the centre of several rows this year after publicly positioning himself against "wokeness".
Earlier this year he launched his own political party, Reclaim, which claims to fight for freedom of speech, aiming to “reclaim British values” and challenge a “culture war” taking place in the UK.
The debate around the playing of the original Fairytale of New York, which also contains the word 'slut', has become something of an annual talking point.
Last year, the Gavin and Stacey Christmas special caused controversy after Rob Brydon and Ruth Jones' characters Bryn and Nessa sang the original complete with the slur.
Following 866 complaints about the episode, with some referring to homophobic language, the BBC said in a statement that "there was no intention to offend viewers".
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