While Nicolaus of Damascus, a Greek historian and philosopher who lived under Rome during the early empire, attributes gladiatorial games to the Etruscans (who inhabited the area a bit North of Rome while the great city was still in its youth), most modern historians agree that they first originated in an area called Campania - where, historically, the first Roman gladiator camps were.

The exact purpose of the 'original,' gladiatorial games can't be pinpointed, although they most likely first found their way in to culture as something called "munera" - essentially, funeral rites. Livy, a Roman historian, wrote that they originated as a celebration of the Campanians, after a successful war with a neighboring tribe called the Samnites (who, it is worth mentioning, have a style of gladiatorial combat named for them).

So, while the Romans themselves technically were not unique in holding gladiatorial games - in fact, they commonly called it a foreign tradition, it was certainly local to the Italian peninsula.

While not a classical culture, it is worth noting the Aztecs had a similar tradition to gladiatorial combat, in which a captured prisoner would be given toy weapons and be pitted against a fully armed warrior. Not necessarily the same thing, but, oh well.