Medium creatures cannot wield Large 2-handed weapons…
…without an ability that says otherwise. The rules on Inappropriately Sized Weapons include
Inappropriately Sized Weapons: A creature can’t make optimum use of a weapon that isn’t properly sized for it. A cumulative –2 penalty applies on attack rolls for each size category of difference between the size of its intended wielder and the size of its actual wielder. If the creature isn’t proficient with the weapon, a –4 nonproficiency penalty also applies.
The measure of how much effort it takes to use a weapon (whether the weapon is designated as a light, one-handed, or two-handed weapon for a particular wielder) is altered by one step for each size category of difference between the wielder’s size and the size of the creature for which the weapon was designed. For example, a Small creature would wield a Medium one-handed weapon as a two-handed weapon. If a weapon’s designation would be changed to something other than light, one-handed, or two-handed by this alteration, the creature can’t wield the weapon at all.
A Large one-handed weapon would be a two-handed weapon with a -2 penalty for a Medium creature (this is often how the iconic Barbarian Amiri is depicted). A Large two-handed weapon would be one weapon size greater and unwieldable.
It is further established (on the weapons page linked above) that throwing a weapon (regardless of if it has the Thrown property) counts as wielding it, with all the benefits and restrictions. It also has a rule applicable to throwing a weapon two-handed.
Thrown Weapons: Daggers, clubs, shortspears, spears, darts, javelins, throwing axes, light hammers, tridents, shuriken, and nets are thrown weapons. The wielder applies his Strength modifier to damage dealt by thrown weapons (except for splash weapons). It is possible to throw a weapon that isn’t designed to be thrown (that is, a melee weapon that doesn’t have a numeric entry in the Range column on Table 6–4), and a character who does so takes a –4 penalty on the attack roll. Throwing a light or one-handed weapon is a standard action, while throwing a two-handed weapon is a full-round action. Regardless of the type of weapon, such an attack scores a threat only on a natural roll of 20 and deals double damage on a critical hit. Such a weapon has a range increment of 10 feet.
There are classes that get around this with archetypes. Notably, the Barbarian Titan Mauler gets the ability to do so at level 3 and the Titan Fighter does so at level 1.
If you wanted to homerule these in to make a cool moment, you could borrow one of those abilities’ penalties and apply some other ‘cost’ (perhaps Fatigue after the fight or Strength damage from a stressed muscle) to the character later to establish that doing what they did was not something they could typically do.