Hiya! So there were originally some comments here that I wanted to respond to before, but have been removed by for some reason. However, strictly speaking, those comments seemed to be devolving more into discussing logistics (acquisition of material, recruiting and arming of men) rather than tactics (battlefield dispositions, movements, and operations). Logistically Rome was very much the gold standard in terms of logistics, but I don't think that technically answers the question and there were some very questionable points even in those logistics answers. So let's cover tactics!! Now strictly speaking, my background is in late-medieval/early modern transitions, so my ability to address issues of classical warfare may be a bit Spartan (no pun intended :D)
However, generally, I think the answer to this question is no, there was no large-scale attempts to adapt Roman military tactics. It's a bit difficult to completely divorce tactics from logistics and technology, and any Roman army would likely greatly outnumber a medieval army. However, this is not the same thing as tactics. Take it at a fundamental level of two evenly sized armies facing each other off. A Roman legion would throw their pilum, adjust their maniples into a front, advance in somewhat tight formations, draw their gladius and stab from behind their shield when in direct contact with their enemy. Keep this in mind, Roman heavy infantry was some of the best the world had ever seen, but skirmishers, light infantry, cavalry, and auxiliaries were seen as just that, auxiliary. By contrast, medieval Europe, was moving towards a more combined or integrated arms methodology in the later period, which would not favor this drastic favoritism towards infantry. Even the so-called "infantry revolution" during the 14th and 15th centuries is perhaps best understood as rebalancing the deployment of the army, rather than the conversion of cavalry to infantry. This, combined with the advent of small arms, would bring about the full military revolution that is often written about in the early modern period.
So what happens in our original scenario? Well the pilum is likely going to fall short or the Roman legion is likely going to have to move forward to obtain range, which will expose them to medieval archers and crossbowmen. There, the blocks of the maniple are going to be exposed to concentrated volleys of arrows and bolts in a tight formation. The Romans favored the infantry advance, which is what they would do, since most medieval commanders favored fixing their infantry as a defensive measure. The medieval infantry would likely be a dismounted knight, who depending on the period would be wielding a polearm with some smaller auxiliary weapons or a pike formation. In either case, their enemy would be wearing full plate armor, which would drastically weaken the effectiveness of the gladius, while the medieval ax, halberd, mace, warhammer, or sword would have no such issues with the Roman armor at that close range. While the infantry is engaging in the melee, the fully armored medieval knight with the latest in saddle and stirrup technology is going to the Roman auxillary cavalry with great ease and then regroup to mount a couched lance charge on the flanks of the Roman legion. The legion may maintain its discipline, it may not, but eventually, simply driving their infantry into the enemy position is going to expose them to the variety of technologically superior medieval armor and weapons.
This is not to knock on the Roman military. As I said at the onset, they were logistically one of the greatest institutions of the pre-modern era, and they were extremely good for their time. But that last part is key, By the end, the Roman legion was gone before Rome really was, and the Eastern Empire adopted completely different tactics to survive, to the point where I doubt the earlier commanders would even recognize the armies of Justinian or Macedonian era. There is some evidence that classical texts were used to create new pike formations in the early modern period, but the wholesale adaptation of Roman military tactics seems unlikely to me, simply because it would involve a complete shift away from the integrated arms approach that was becoming highly successful towards the end of the medieval era. I hope I haven't oversimplified the entire centuries long history of the Roman military, but hopefully if I have, a classicist will be able to straighten me out. But overall, I think the course of the military progression was heading in too different a direction to make adopting Roman tactics plausible en masse.
I hope this helps! Let me know if this is too tactics heavy for you, if you want additional info/sources, or more follow-up!!
Sources (not everything is used explicitly here, but give excellent overviews of the period as well as concepts such as the rise of infantry, integrate arms, etc.):
The Renaissance at War by Thomas Arnold, 2006
Arms, Armies and Fortifications in the Hundred Years War edited by Anne Curry and Michael Hughes, 1994
The Military Campaigns of the Wars of the Roses by Philip Haigh, 1995
Medieval Warfare: A History by Maurice Keen, 1999
Warfare in the Medieval World by Brian Carey, 2006
Warfare in the Ancient World by Brian Carey, 2006
The Roman Army at War 100 BC to 200 AD by Adrian Goldsworthy