So, generally not. The typical dress of a Roman man was an inner tunic, which stretched to just above the knee and could be pulled up further with a belt, with an outer tunic over the top. A toga, more formal dress, would go over both. Women were similar, with two tunics and a palla or more formally, stola over the top. The belt would be used to pull up the tunic when one was out and about, but it might hang loose at leisure or at work. Revealingly, one who was just wearing the inner tunic, as one might in the evening at home or working hard in the field, was known as nudus, which translates just how it sounds.

Underwear did exist, in the form of a loincloth known as subligaculum, which was basically a bandage wrapped about the pelvic area. It was associated with gymnastic exercise, actors, labourers and slaves. Greeks didn’t even wear the loincloth when doing their gymnastics. It wasn’t really the sort of thing a respectable person would walk about wearing, much like trousers.

Much like civilians, the typical Mediterranean soldier’s dress was a short-sleeve, knee-length tunic, a bit longer than the type worn by civilians and stretching down to around the calves, but gathered up by a belt so it hung around the knee. With so much extra material gathered up by the belt, I would imagine it could function almost as an undergarment as well and provide a bit of protection from the chafe and heat that soldiers must have experienced at times (Goldsworthy 2003).

In regards to the subligaculum however, a letter to a soldier in northern Britain (from the Vindolanda tablets, c. 1st and 2nd century BC) shows him being sent two sets of subligacula, showing that at least in this period, Roman soldiers were beginning to adapt their clothing to the climate. This is also the context in which some infantry and cavalry began to wear breeches, socks, and even trousers in certain cold climates (Goldsworthy 2003).

Basically, it wasn’t necessary to wear underwear in the Mediterranean climate. If one was cold, one could simply wear more tunics, which is what the Romans typically did (Augustus wore five layers in winter). Just keep in mind that the tunics could be a bit longer than necessary, so as to be gathered up by a belt, which would give one some extra material and folds to work with down below if one wished for it.

References:

Mostly I’ve referred to Smith, 1875, A Dictionary of Greek and Roman Antiquities (from lacuscurtius.org). Also see Goldsworthy, 2003, The Complete Roman Army, which has a short section on soldier’s dress.