Depends on sources and levels, but advantage: monk.
It’s going to depend a lot on what sources are available, and how exactly we define “fighting unarmed.” Also, what level are we talking?
Everything on the table
Wild monk
Assuming we allow all official content—read: Dragon magazine—and accept any kind of fighting unarmed—read: not just unarmed strike—and are talking about 20th level, monk wins hands down because of the variant in Dragon vol. 324 that gets wild shape. It’s not as good as a druid’s (fewer uses per day, slower progression to larger and/or elemental forms), but it is something we can actually value more monk levels for, which is more than we can say of fighter levels (or other, non-wild monk levels).
Even if we do stick to unarmed strike, wild shape is probably still the best approach available among these two classes for focusing on it. Larger sizes and massive ability score bonuses go a long way here. The monk’s scaling unarmed strike damage also combines very nicely with those larger sizes—at 16th-level when Huge sizes become available and the monk has a 2d8 base unarmed strike damage at Medium, you’re looking at 4d8 base damage on the unarmed strike. (Awkwardly, at 20th level, the monk’s Medium damage improves to 2d10—which isn’t an entry on the weapon size table and we have no official indication of how that damage scales to larger sizes. Extrapolating from the 4d8 of the Large monk, though, it becomes fairly clear that it should be 6d8, based on the 4d8 of a Gargantuan bastard sword upgraded one size to 6d8.)
Wild shape may also be the only way these two classes have of accessing pounce while unarmed. Obviously the best approach is to just wild shape into something with pounce, but there are also some feat options for it that require wild shape: the Lion’s Pounce wild feat from Complete Divine lets you pounce by burning a use of wild shape, while the Bestial Charge tactical feat from Complete Champion lets you pounce on the round after using wild shape (or the same round, if you have the ability to wild shape fast enough). These are both very limited, but pounce is also very good. Still probably better off just wild shaping into a shape with pounce, though.
Wild shape is also by far the best way of grappling things yourself (as opposed to summoning grapplers, which is even better). Grappling is also something that’s fairly effective, ish, and its damage is based on your unarmed strike damage. So that’s nice.
Huge fighter, huge LA, huge dungeoncrash
To keep up, the fighter, by contrast, will probably want to take some higher-LA race to get larger—which their HD and BAB does allow them to afford, to an extent, but only so much and it always hurts. Large size at LA +1 is not hard (half-minotaur is popular), but larger is painful. Creating a half-goristro template (based on the goristro from Fiendish Codex I and these rules for creating half-fiends out of specific types of fiend) could get it at LA +4, but LA +4 is awful, and this source is, if anything, even more dubious than Dragon magazine. An alternative might be a half-minotaur goliath, who would be Large and have powerful build at LA +2, getting many of the benefits of being Huge (but not the space or reach, the latter of which is especially valuable).
The LA causes the fighter to give up most of their advantage in HD and BAB, since LA gives neither. However, they have the option of the dungeoncrasher variant from Dungeonscape, which allows them to deal large damage every time they bull rush a foe into a wall. The Knockback feat from Races of Stone lets them do that on every attack. The pugilist variant from Dragon 310 can get them Improved Unarmed Strike (and a base 1d4 damage with it) and Endurance for free (read: in addition to a 1st-level bonus feat), which is nice. Superior Unarmed Strike from Tome of Battle can give them scaling unarmed strike damage, though not as good as a monk’s.
The fighter probably can’t get pounce (Races of the Wild’s Catfolk Pounce or Player’s Handbook II’s Two-Weapon Pounce are conceivable, but both are mediocre, limited versions of pounce and very costly). However, the fighter can take Travel Devotion from Complete Champion, and can afford it more easily than a monk can—which is good, because the fighter probably needs to take it 3 or 4 times. Pounce and Travel Devotion each have their pros and cons, but here Travel Devotion is a con primarily because the fighter would really like to charge, but without pounce probably can’t. Travel Devotion also requires swift actions, which is annoying because the fighter would like to have a swift action available to zap a wand of blockade to make sure they can dungeoncrash. Flying arguably eliminates that concern, though.
Comparison
Ultimately, there is just so much that wild shape allows access to. Yes, the fighter can get the size by taking on LA—and even manage it earlier than the monk can, allowing them to possibly be Huge before the monk even has wild shape. But the LA also makes them preposterously vulnerable at low levels, so it’s not really viable until mid-levels anyway. At that point, wild shape is looking very competitive even if you don’t have Huge size yet. Once you get it at 16th, the monk is clearly superior.
For some numbers on the viability issue, an ECL 5th half-goristro has, at best, something like 17 or 18 hp—any actual threats at this level are going to kill the fighter as an afterthought. The monk has at least 32 hp, and since they’re going for wild shape (and won’t need their own Strength or Dexterity after 6th), probably more like 37 or 42. The fighter gains more hp per level, though, and pulls ahead at 10th or so. So you’re looking at a sweet spot between roughly ECL 10th and 15th where the fighter is maybe competitive, but above that or much below it, the monk looks far better.
But even in the sweet spot, the fighter is giving up a lot of their advantage because they have to take Travel Devotion multiple times, and they have fewer actual levels, and monk also gets some bonus feats, so the fighter might actually have fewer feats. Meanwhile, wild shape just... covers a multitude of sins. So even in the fighter’s sweet spot, the monk is probably better.
Source or style limitations can change the outcome drastically
If we cut Dragon magazine, the fighter can still dungeoncrash but the monk can’t wild shape. If we leave Dragon magazine, but decide wild shape isn’t in keeping with the theme, then the fighter is even better off, while the monk still can’t wild shape. The monk also finds the LA that the fighter is taking on much more painful to acquire, though without Dragon or web articles the fighter is stuck with probably goliath (which Knockback lists explicitly as qualifying even though it doesn’t seem to?).
On the other hand, the fighter has fairly limited options outside of dungeoncrasher, and dungeoncrasher might be considered a little outside the theme, too—yeah, you’re making unarmed strikes, but basically all the damage is coming from the dungeoncrash effect, and your unarmed strikes just happen to be the weapon you’re using while triggering Knockback. If dungeoncrasher is nixed for theme, monk can do a lot better even without wild shape, thanks to stuff like invisible fist from Exemplars of Evil and the honestly-kinda-broken dark moon disciple from the Champions of Valor web enhancement (at 7th level, in anything less than full daylight, or daylight, you get permanent total concealment, which makes you un-target-able). Though these advantages themselves have little to do with unarmed attacking, and so may not count in the competition.
If you’re literally just looking at unarmed strike expected damage, the fighter probably wins. Superior Unarmed Strike means the fighter is two die sizes below the monk, but without size increases that doesn’t matter very much, and the BAB and bonus feats of the fighter allow for much better unarmed strikes overall. Flurry of blows is good, obviously, though the fighter somewhat mitigates the advantage with iteratives (though flurry bonus attacks are at your highest BAB instead of at increasing attack penalties, plus eventually you get two of them).
