yes and no (although i voted yes)
many factors = "grunt" but if all else is equal, then yes, hitting your speakers with more watts tends to change the tone into a "fuller" "fatter" tone with some "grunt". For a long time I've noted that using less watts per speaker results in a more transparent and articulate tone, while using more watts equals a fuller fatter tone.
I like both tones and lean towards one or the other at any given time. Sometimes i bring a bunch of cabinets and go for the articulate tone, other times i beat the heck out of one or two cabs for the "grunt". Some have said that the more watts approach is inducing a level of distortion, which is probably true - the sonic charactaristics of the speaker changing indicate that, but if that's the coloration you;re after, it's all good.
As far as other factors go, they are leigon. The way you play affects tone- a harder attack obviously can get more "grunt", right hand position can influece bass response, etc. The bass itself will dictate a good amount of the tone. The voicing of your preamp also dictates a good % of the tone as well as the voicing of the EQ and how you set it. Tube power sections might not = Bass response, but some sure add heft and "grunt". However, in terms of the final sound, the voicing of the speaker cabinets themselves are probably the most important link in the chain. Some cabs have "grunt" and some don't. some have huge bass, some don't. IME, if they don't have it, you really can't EQ it in very well. And yes, bill is on the $$ (although i think watt do mean something if all else is equal), a cab with low sensitivity and a crappy frequency response will not help you out much.