They certainly did. The caveat lector here is that customs of what is inscribed on tombstones do change a lot over time and space. Elaborate inscriptions on tombstones are not very common anymore today, but another place and time where they were was the Roman Empire.

One of the most prominent examples is this tombstone of a legionary centurio of the 18th legion, Marcus Caelius, erected in Xanten, once a garrison town of the Empire. In the third line from the bottom, we read: "cecidit bello Variano", who fell in the war of Varus (the great Roan defeat at Teutoburg forest).

Other tombstones mention sickness, drowning or even medical malpractice as causes of death. It wasn't really usual to give the cause of death on the tombstone, but it wasn't unheard of, either - Roman tombstones used to be quite elaborate, and, especially when made for close family members or loved ones, people often gave more information about the deceased than was expected, such as cause of death.