We don't know.

Sponsalia (think formal engagement ceremony) rings go back to republican Rome.

Archaeologists (and looters and the like) have found a load of Roman era rings that express various romantic and marriage sentiments but there's no documentation. Could be wedding rings, could be betrothal could be more general love tokens. There is a subset of Roman rings that have a bezel with male and female heads facing each other on them and its been argue that at least some of these constitute marriage rings.

If you want a latest possible date then Isidore of Seville writing in the 7th century mentions the spouser giving a ring to the spoused which is worn on the fourth finger. Which pretty unambitiously describes a wedding ring.

sources

The Jewellery of Roman Britain Celtic and Classical Traditions by Catherine Johns

Rings: Symbols of Wealth, Power and Affection by Diana Scarisbrick