Can I use mulled wine for deglazing a beef stew?
Of course you can use it.  You can use any edible liquid:  water, wine, chicken stock, pineapple juice, whatever.

The real question is whether the flavors the wine was mulled with are compatible, and you enjoy them.  If they will compliment your dish, go for it.  I cannot speak to your taste in stew but the sweet spices often used in mulled wine may give it a Mediteranian or middle-eastern feel--or they may be too faint to detect.

Contrariwise, you probably will do just as well with plain water, or if you are so inspired, vodka. 

One reason (among many) that deglazing or cooking with wine or spirits is popular is that some flavorful compounds are not soluble in water or fat, but are soluble in alcohol (which, contrary to myth, does not all "cook off")--and so the residual alcohol from the wine can synergistically make the dish seem more flavorful.  This is the science behind the vodka sauce fad, for example.  

Its stew--its not haute cuisine :-)  I say go for it, either way, and enjoy.