No matter the half-orc monk's level, the typical half-orc monk that possesses Strength 18 that's using only material from the core rules can't throw a pig.
Step 1: Find the pig
Officially, pigs have no statistics. The Player's Handbook lists on Table 7–3: Trade Goods lists one pig as an item with a cost of 3 gp (112), but, otherwise, the entire Wizards of the Coast Dungeons & Dragons, Third Edition corpus, inclusive, is silent on the statistics of the pig. Pigs are, apparently, window-dressing, a creature like chickens that the DM includes in the background. "The pig is unimportant, okay?" says the weary DM to the half-orc monk's player. "Stop trying to throw it, and get back to the adventure. Please?"
However, straying beyond Wizards of the Coast material leads to the officially licensed Kingdoms of Kalamar product Dangerous Denizens that has a pig entry that says, "For the statistics of an average pig, use the boar entry from the Monster Manual [on page 270 of that book and here], replacing the gore attack with a bite attack doing equal damage" (191). This makes a pig extremely dangerous, as should be true of a full-sized adult pig: "[I]n the USA and Canada [as of 2016] approximately 40 people are killed each year by pigs[; that's] six times more than by sharks worldwide" (source).
Because it's a more interesting question if there are actual pigs in the foreground rather than green-screened pigs in the background, the remainder of this answer assumes that a pig is the Kalamari pig-is-a-boar-with-a-bite-not-a-gore kind of animal.
Intermission: Wearing magic slippers to throw pigs
Tome of Battle has the martial discipline Setting Sun (70–5), and many of the discipline's maneuvers, if successful, see a martial adept toss a foe (like an enemy pig) anywhere from 10 ft. to more than 60 ft. away from the martial adept—with no need to grapple the pig and no need for the pig to be helpless.
While a half-orc monk probably does himself no injury by picking up a few maneuvers with the feat Martial Study (31–2), a better option is the magic item slippers of the Setting Sun (150) (3,000+ gp; 0.5 lbs.), perhaps using the Magic Item Compendium's rules on Improving Magic Items (233) to add to the slippers more and more slippers so as to meet the prerequisites of high-level Setting Sun maneuvers without taking feats or levels of martial adept classes. (Ask the DM first, though, if combining multiple pairs of slippers into one pair of slippers is legit in that DM's campaign; it's a little dubious.)
Step 2: Hold the pig
Even at level 1, a half-orc monk that possesses Strength 18 is already even on his grapple checks against a pig. However, without the feat Improved Grapple (PH 95–6), every time he tries to grapple that pig, he'll provoke an attack of opportunity from the pig, and a level 1 half-orc monk can be downed by one good hit from a pig: the pig's bite deals the same damage as a longsword wielded by a Medium creature that possesses Strength 16! The lesson? Either the monk should first render the pig helpless or the monks should take the feat Improved Grapple—it's available to a level 2 monk as a bonus feat so that the swole half-orc needn't to meet the feat's stupid Dexterity 13 prerequisite—or the half-orc should befriend the pig sufficient that it's agreeable to the monk's wacky plan. Then the monk can pick up the pig.
This writer assumes that the reader knows how to render a pig unconscious. (Hint: It involves sticks. And hitting.) This is the path of least resistance on the way to pig-tossing town. And, as mentioned earlier, grappling is a bad idea unless the monk is specced for it or has a hp or AC sufficient to weather a few pig blows, but if the monk is specced for it or if the monk is far more powerful than a Challenge Rating 2 pig, then the pig's probably grappled. (I suggest the monk wait until level 15 or higher before attempting to engage the CR 2 pig because, y'know, he's a monk.)
If the monk possesses at least 1 rank in the skill Handle Animal, the half-orc could take a full-round action to employ against the pig the skill Handle Animal to "push" the animal to perform a trick it doesn't know (DC 25), picking the trick play dead. The pig doesn't know this trick because this writer just invented it. (For the game not in real life.) The Handle Animal skill description, in part, says, "Possible tricks include, but are not necessarily limited to," the ones found in the Player's Handbook and, presumably, elsewhere, too (74), therefore allowing—with the DM's permission—pretty much any trick, and, really, the trick play dead shouldn't be a hard sell. A successful Handle Animal check in this regard should allow the half-orc monk to pick up the pig. (A creature that doesn't possess at least 1 rank in the skill Handle Animal can make against a domesticated pig a Charisma ability check (DC 25) to get it to do this. Good luck!)
Step 3: Hurl the pig
If the pig has been rendered helpless—through pummeling or persuasion—, this reader believes it's best to treat the pig as an object. (That is, this DM—who does his best to play by the rules—doesn't require PCs to grapple helpless PCs then require the PCs to continue to use the grapple rules to haul those helpless PCs from place to place. The game's gotta eventually bend to common sense, for verisimilitude if nothing else.) However, the game has no rules for throwing objects just to see how far (or, for that matter, how fast) a creature can throw an object.
Instead, the game has rules for throwing actual and improvised weapons. (Yes, a great many sports in D&D 3.5 are impossible to simulate without house rules.) Treating the pig as an improvised thrown weapon sees it have a range increment of 10 ft. with a maximum of 5 range increments. To throw the pig at a square 50 ft. away and hit that square, the half-orc monk makes a ranged attack roll against AC 5. The monk on this attack roll suffers a −8 penalty due to range and −4 penalty due to nonproficiency. (Proficiency with the pig is not impossible to attain but beyond this answer's scope.)
If the half-orc monk's attack roll misses, this DM would determine the pig's actual landing square as if the pig were a a splash weapon (PH 158); this gives a 5-in-8 chance of the monk having accidentally hurled the pig farther away than he'd intended although not to where he intended.
(A thrown weapon's range increment can be increased by, for example, the feat Far Shot (PH 94) that doubles it. Whether a half-orc monk is willing to commit valuable feats solely to tossing pigs farther will depend mightily on the campaign.)
If the monk has grappled the enemy pig, then, like this fine answer says, typically other game elements must be marshaled to throw the pig. The lone published-in-a-book method is the feat Fling Enemy (Races of Stone 140), but it's enormously complicated for a half-orc monk to meet the prerequisites of the feat: Strength 19 isn't 18, the half-orc monk lacks the specifically racial ability to throw rocks therefore he requires the feat Rock Hurling (143), and he must somehow be at least size Large. This is, in almost every case, wholly unreasonable, consuming resources significantly beyond its final benefit. In short, while a half-orc monk can probably find a way to take the feat Fling Opponent, he probably shouldn't, contenting himself with squeezing the pig into unconsciousness first then throwing it as described above.
Alternatively—and well beyond Races of Stone—, a generous DM may allow the half-orc monk that grapples then pins the pig to exercise from the Rules of the Game Web column "All about Grappling (Part Three)" the grapple option Toss Your Foe:
[You m]ake an opposed grapple check as a melee attack. If you succeed, you can literally pick up your foe (provided you can lift your foe's weight). Make a Strength check; if your result is at least 10, you toss your foe 5 feet. For every 5 points your Strength check result exceeds 10, you toss your foe another 5 feet, to a maximum of 25 feet.
This is pretty clear, simple, and direct. Unfortunately, despite the columns sometimes being the only source of insight into design decisions and of caulk for rules gaps, the Rules of the Game Web columns have reputation that's less than stellar (see here).
Note: To wield the pig as a melee weapon, see this question.