Everyone in these balance discussions seems to talk as if the more powerful class has every single relevant bonus maxed. The rogue has maxed all their class skills, the wizard has every single spell at their fingertips plus all the item creation and metamagic feats they want, the cleric has ten rounds to buff himself for the big fight, etc. etc.
I don't see it as an unreasonable assumption. The rogue has 8+int mod skill points every level, what would he max, if not his core competencies? Knowledge skills?
I don't see where it assumes that the wizard has every spell at his disposal. Ditto for the cleric buffing. Typically, at higher lvs, quickened divine favour+divine power will suffice. The remaining buffs are nice, but not needed.
Just why can a fighter not do it? Only because the rogue exists in some way to make the rogue feel special?
There is nothing stopping the fighter from trying to pick the merchant's pocket. But the DC will surely be so high (if it was low, why didn't the rogue with maxed out ranks in sleight of hand do it?) that he cannot succeed on the check, even on a natural 20. And the party will probably be worse off as a result of his botched check.
This also brings me to the next point. The rogue will surely have higher ranks in sleight of hand than the fighter. Any time a check is required, the rogue is the obvious candidate to attempt it. I cannot imagine why you would ever want to the fighter to make the check instead of the rogue.
So, can a fighter pickpocket? Sure.
Should he be the one to attempt it? Obviously not! It is like saying if I can punch a policeman in real-life simply because I don't like his face. Sure - there just going to be a long jail term in my future.
It is really an issue of identifying your core competencies, and focusing on them. The fighter is no rogue, so it there is little point in him trying to emulate one.