Rome is a really interesting case to look at medicine in the military, because it's an important time in history where battlefield medicine actually pushed civilian medicine forward rather than the other way around. (Clarification upon request: overall, not just in terms of emergency medicine and wound treatment--things like preventative health and even, it seems, social attitudes towards physicians).

We do have to be careful to distinguish time periods within "ancient Rome." In Man and Wound in the Ancient World, Richard Gabriel points to a few key turning points in Roman history that spurred advances in medical treatment for soldiers, both during and outside of battle. This refers both to organization/personnel and to medical knowledge/skills.

For the Republican period as a whole, battle medicine seems to have been left up to individual people. John Scarborough points to influence from the Greek world that pushed Roman military commanders to hire and pay their own personal medical officers. How far down into the legion that care extended, had to depend on the individual commander. There are a lot of references in sources to soldiers treating themselves, but also to the occasional commander like Publius Scipio seeing to it that the physician treating his wounds also treated his soldiers'.

A major jump forward comes before and during the Gallic Wars, so around the time of Caesar. Our good friend Cicero indicates that soldiers could probably expect a medic to come treat their wounds, which indicates a step forward in terms of organizational sophistication:

Moreover, we often see, when the wounded are carried off the field, the raw, untried soldier, though but slightly wounded, cries out most shamefully; but the more brave, experienced veteran only inquires for some one to dress his wounds

If this seems like a tangent, it's not quite: you'll notice Cicero's very classically Stoic perspective on gutting up, dealing with wounds, and carrying on. Would a proper Roman soldier let himself be stopped by a case of the sniffles?

Perhaps not, but he didn't always have a choice! Julius Caesar himself indicates this in Gallic War, VI:

Sextius Baculus, who had led a principal century under Caesar (of whom we have made mention in previous engagements), had been left an invalid in the garrison, and had now been five days without food. He, distrusting his own safety and that of all, goes forth from his tent unarmed; he sees that the enemy are close at hand and that the matter is in the utmost danger; he snatches arms from those nearest, and stations himself at the gate.

(ETA: See /u/TangosWithNapalm's remarks below for a deeper analysis of this passage.)

At this period in time, disease among soldiers was hot on the consciousness of military commanders. Around then and through to the civil wars following Caesar's assassination, apparently diseases ravaged the ranks even more than usual. This helped push the next major innovation in Roman battlefield medicine: attention to the prevention of sickness.

By the full blossoming of the imperial period, especially in the first centuries CE, Roman military medicine was amazingly sophisticated for its day in terms of professionalization and official support, preventative health, and wound care--the sooner the better. Sanitation, exercise, and good diet were seen as vital to ensuring the most fit troops possible. Roy Davies even describes forts that had indoor exercise areas for maintaining fitness in bad weather. This focus on preventative medicine shows an emphasis on entering battle with the best, healthiest, fittest army possible.

And when things went wrong, the Roman Empire was not about to throw away its valuables. One of the key planks of the newly professionalized military medical service was the inclusion of a hospital or convalescence area in military forts. We might see this as an institutionalization of what Caesar referenced: a place where sick or injured soldiers had time to heal, under professional direction, rather than being left to the chancey abilities (and existence...) of a local civilian population.

So while the Stoic underpinnings of the Roman worldview led to poetic proclamations and probably an underlying valuation on sucking it up, at least by the time of the Roman Empire, the military thought it best to go into battle with hearty and hale troops. Although direct references are scanty, and surely the line between "sickness" and "health" varied incredibly from person to person and needs-of-battle to needs-of-battle, there was definitely a point where Romans recognized it was better for everyone concerned to stay behind.

This is a great question, follow up - would soldiers ever fake being sick to get out of duties/battle?

If I may add another follow up, how organized and/or capable were field medics? Did the romans even have them

Vegetius wrote on this particular aspect of soldiers' health as part of his military treatise De Re Militari, which covered the importance of military logistics, including supply lines and ensuring that soldiers had access to clean drinking water.

There are several points in the text (Book II) at which Vegetius describes the manner in which officers can perform substitute duties for officers who may be sick in need of replacement, which implies that those officers would be removed from their duties in the event of illness, but this portion of the text refers specifically to the duty of specific ranks of officers, not common footsoldiers. There's also specific portion in the text where Vegetius says that sick soldiers should be brought back to health with proper feeding under the care of doctors.

However, it's important to note that De Re Militari is that it is mostly a prescriptive treatise by Vegetius articulating the manner in which the legion ought to be run, rather than a record dedicated to documenting how things were actually run at the time Vegetius was around to observe them. In fact, according to the introduction of the 1940 reprint of Lieutenant John Clarke, De Re Militari "was little used by the Romans for whom it was written."

To provide context for Vegetius' work, De Re Militari alludes to events that took place in 383 (the death of the Emperor Gratian), meaning it was written and published some time after, and this was a time at which the decline of the Roman Empire was already on the horizon, in the form of the civil war that led to Gratian's death, as well as invasions throughout the late 4th century (e.g. Germanic invasions in the 370's). The first book of Re De Militari is essentially a call for military reform, meaning the things that Vegetius called for (access to clean drinking water for soldiers, proper medical care from doctors) were shouted amidst the death knells of the Roman Empire. The document was intended to eulogize the Roman legion opining about its better years, but Vegetius can't be considered a primary source on the "glory years" of the Roman Empire during the first century more than 300 years before his treatise was published.

To also provide appropriate context for the author himself, De Re Militari is one of the few military "manuals" from the Roman Empire to have survived to the present day, but its value is questionable according to some historians; according to G.R. Watson in "The Roman Soldier," Vegetius was "neither a historian nor a soldier: his work is a compilation carelessly constructed from material of all ages, a congeries of inconsistencies," so there remains the question of how much of Re De Militari actually describes the Roman legion as it once functioned, as opposed to Vegetius' idealized image of it.

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