This is quite a vague question, but I'm going to interpret it as follows. Vietnam is not by any means my area of expertise, but I think it can be defined as a collective cultural trauma in the United States of America. While comparisons like this often lead to anachronismic faults, the question wether Rome has or hasn't gone trough a similarly impactful trauma is an interesting one.

One collective Roman trauma I would call greatly impactful in a military sense, are the wars of the Romans against the Germanic Cimbri and Teutones (113-101). In particular the Battle of Arausio (Orange), where the Romans suffered defeat, has been increasingly impactful in the years to come. It is often said that these events eventually led to the Roman tumultus (a state of crisis).

The Battle of the Teutoborg (9 CE) forest needs to be mentioned here as well, at is if often credited as being one of the most impactful collective traumas of the Roman Empire. It was literally named the 'Varian Disaster (Clades Variana)' by various Roman historians.

It is quite difficult to say wether the impact of these events 'match' the impact Vietnam has had on American culture, but it most likely changed the decisionmaking of the Roman Empire when it came to going to war - much like the impact the Vietnam loss had on the United States.

Sources:
Tacitus, Annals IV.66
B. Kerremans, Metus Gallicus, tumultus Cimbricus? The possible promulgation of a tumultus in the Cimbrian War.
Velleius 2,118.