There are frequent critics of realism in RPGs, which point out that RPGs do not and cannot produce fully realistic results. There are often glaringly unrealistic results, and the results that there are usually have very few inputs (i.e. a character stat, a few modifiers, and a die roll). Based on this, they conclude that attempts at realistic games are wasted effort.
I feel that this misses the point. Simulations are used in many fields other than RPGs, and many find use for them even with glaring problems. For example, militaries have for over a century used wargames (such as Harpoon by the Navy) as a means of teaching officers about tactics. The wargames used are not necessarily particularly more realistic overall than RPGs. They frequently make very gross simplifications of damage, for example. However, they are thoroughly tested so that they demonstrate basic tactics.
This might seem far-fetched as a goal in RPGs, but I will offer a personal testimonial. As a child (in the 10 to 15 range), I played the original Traveller science fiction RPG released by GDW. This has a detailed system of generating random star systems: including the star's stellar rating and luminosity, the planet's position, orbital eccentricity and axial tilt, and even the planet's albedo. This was all real science -- and explained it in a manner which is far more interesting and absorbing than some dry textbook. Since then, I have gone on to get a PhD in Physics from Columbia University and work on an international cosmic ray experiment.
It is often cited that realism inherently makes a game more complicated and less playable. There is some truth to this, but I would say that it is a fallacy that more variables are required for more realism. It often aids realism to abstract away detail. For example, you might have real-world statistics on the chance of fatality for different types of gunshot wound -- but no information differentiating the chances by hit location. In this case, it would be less realistic to add in hit location, because once you specify hit location then you no longer have data. It is more realistic to keep the system at a lower level of detail.
As another example, I am trying to designing Wealth rules for my Vinland game, and I would like them to be realistic. However, my sources are limited to archeological speculation and the sagas from the period. Thus, I have no need or want to detail every transaction. Since I don't have price information, assigning specific prices would almost certainly make my system less realistic. What I want to get correct is basic principles like how much of a burden weregeld is, a rough impression of social mobility, and so forth.
I would point to roughly three categories of realism in RPGs: scientific, historical, and tactical. Tabletop RPGs are not a very large field, so there are fairly few examples of what I would consider realistic RPGs in any of the categories. I have a fairly broad knowledge of published RPGs, but I may miss some.
Here, "scientific" basically means physics and astronomy. I do not know of any RPGs which cover biology, chemistry, or other subjects well. Note that there are dozens of science-fiction RPGs. However, many of them freely mix science and fantasy, while others are well-researched but skim over actual calculations and details. I should note that many RPGs (more than can be listed) are good at exercising basic math and probabilities.
RPGs are actually a very interesting approach to history. Traditional history lessons are macro-histories, where students learn sequences of kings or presidents, dates of wars, and so forth. RPGs offer instead a micro-history approach, where the players instead get a more complete picture of what life was like at a particular time and place. This is the approach taken by living history museums and historical recreation organizations. The problem is that there are fairly few straight historical RPGs. Even if a game is written by a well-read author, if play itself is highly ahistorical then the game conveys little of the period.
I actually know very little about military tactics and strategy, so I cannot actually recommend any games. I have the impression that Millenium's End from Chameleon Eclectic has a fair handling of modern firearms. Also, The Riddle of Steel from Driftwood Publishing supposedly has a good treatment of European Renaissance-era fighting styles. However, this is strictly rumor.