(This is a long one so grab some water)
Speaking as someone who was in the U.S. Army, and on top of that, in a unit that had women serving in a front line combat capacity (MP Combat Battalion, every female soldier, and male for that matter, in my Company had a CAB) I can say, at least in "Modern" FPS's (Call of Duty and it's ilk) that women have absolutely no place as a player character.
Fact of the matter is, almost all of these modern setting FPS's involve the player being part of, or something there equivalent to, a Special Operations Troop. There are no special operations troops in the world that employ women in Combat roles. That's not conjecture, that's a fact. Women DO serve in U.S. Army Special Forces, MARSOC, NAVSPECGRU, as well as many foreign Special Operations Units such as English SAS and SBS, however only in clerical or administrative positions. Even the Russian military, who has historically been the most open to allowing women to serve in "traditional" male roles (Russian being one of the few countries to have multiple female fighter Ace's, EXTREMELY proficient snipers, and female chiefs of staff) To put it simply, women do not kick down doors in Special Operations. And, for the foreseeable future, they won't.
There are a variety of reasons, some range from as innocuous as ability for a special operations troop to blend in with their surrounding (a female soldier would stand out quite a bit more in Kandahar trying to gain intelligence from a local clan) to as extreme as WHY women are not allowed in front line infantry units. It is funny, even within the military it is not... often explained *why* women are NOT allowed in true front line units (In the Army's case that would be Combat MOS's in Combat Arm's branches (Infantry, Armor, ect). As far as the U.S. Military (and English Military although they have an extra reason) is concerned there are two critical reasons why women are prohibited from these positions:
1) Women, baring the rare exceptions, have grossly different physical and hygienic demands that are not conducive to a front line Infantry unit. If a male soldier goes 3 months without showering and at best changing/washing clothes once a week, in all likelihood he will be physically fine. Probably not happy, nor anyone catching a whiff of him, but he will not develop (baring open wounds or something of that nature) an infection PURELY from not showering/changing clothes (I can speak from personal experience on two tours on this). A female, without being overly graphic, cannot. They will develop specific infections that are unique to their gender due to lack of hygiene. Hygiene is important in the field, but to many degree's it goes by the wayside.
Additionally women, baring the rare exception, can't meet the same physical requirement. That's a data supported fact in the U.S. Military and the English military. Women even have separate physical standards in the U.S. Military that are VERY much different from the males (for example, on the Army APFT in the 17-21 age range, the female 2Mile run time of 15:36 is a 100% score, while the same time for a male equals a 64, just 5 points shy of failing the exam, what this is in essence saying is the absolute best score time for a woman in the Army is just BARELY passing for a male) These score discrepancies don't exist to make it easier for women, nor do women pass the APFT in a far greater percentage. In actuality, the pass/fail ratio is relatively similar between both genders. Women simply are not as strong as men. Period. We're different, what a surprise. However that degree of difference, while not a problem in everyday life, or even clerical/administrative duties in the military, both are a far cry from front line service such as infantry. NOW take it to the next point, given this is about games and games are almost always about Special Operations, Special Operations universally have DRASTICALLY higher physical requirements than any regular military. They in fact are SO high, that barely 1% (was like .96% or something last time I saw the statistic) of the U.S. Army, male or female, can consistently make the standard.
So in summation of that, not only can women not meet the "bare" minimum of standard physical requirements, there is in no way shape or form, that they could meet the physical requirements of a Special Operations troop.
2) This is the most unknown, but not hidden though, reason, and that is physiological, and not in the way you'd think. this ALSO is the reason there was resistance to repealing don't ask don't tell. Why no 4 star ever mentioned it in the press is beyond me as it is a fairly... logical reason to bar both homosexuals and women at least in frontline combat units.
Men have a natural urge to protect. This is a biological fact. Those rare individuals that don't are the exception, not the rule. In the military, this holds true. However the urge to protect a battle buddy and perhaps, someone you're attract too, are drastically different. In the military there is one thing held above all else; the mission. Absolutely nothing can come between the soldier and the mission. No danger, no person. Only a threat to the mission itself can cause pause. This is paramount and when you become an officer this is drilled home all the more clear. Personal relationships CANNOT be allowed to come between the soldier and the mission.
What this means is a soldier cannot allow himself to be in a position where they choose to save a friend/romantic interest over completing a mission. Period. End of story. Black and white (in this rare time, the world is no gray). Now a group of straight men, or for that matter, a group of straight women, this problem won't arise, however if you mix units, or have homosexuals involved, then a problem can ensue. We've all been attracted to SOMEONE at a job or school before. Can anyone honestly say that did not effect their work behavior and perhaps even how hard they worked a company or school goal? Even something as simple of skipping a class to be around that individual or rushed paperwork at a job to finish early or took an extra long lunch break, or even just being distracted. While... understandable (and even perhaps tolerable) in the regular world, this is absolutely UNACCEPTABLE in the military. As an MP, and our unit being COED, I saw how this *did* cause regular issues, both in and out of the field. It is a TRUE problem and MP's are relatively small units. I can only image the problems that could happen in a large Infantry battalion.
You couple this with the scientific fact that men have an urge to protect women, and there is a spell for disaster. To return to the idea of all women units, that in theory could work 100% fine in say a regular infantry unit, but keeping in mind the overarching theme here of women in the military IN games, this does not work. As said, even few MALES can attain the necessary physical skills, and qualifications to obtain Special Operations quality, and granting that the exceptional female may have that ability, there is not enough of either to attempt to have an all female special operations troop. The Russians tried that once with the NKVD, and I suppose the Amazonian guard might qualify (although I would question whether or not they were as qualified as say, a member of the British SAS, or German KSK)
So if you followed me through this, (long read I apologize) those are the two key reasons right now women are not allowed in "simple" infantry units. To make the next step to Special Operations is just impossible in the world as we know it now. Drastic technology or necessity is all that could change that at this point.
Now to tie it all into the grander topic, in "modern" setting FPS's, women cannot be player characters. At least in any section involving the "classic" idea of what a modern FPS is suppose to be (kicking down doors, setting charges, special insertions/infiltrations)
That said you *could* have women in a modern setting FPS however it would not be like any other FPS as it would have to involve being part of a VERY different military group than Special Operations. For example in the case of the U.S. Military, for all intents and purposes (baring a few, isolated exceptions) the only roles women can take on that also tend to involve "true" combat are MP's (and their Air Force, Marine, and Naval equivalents) SeaBee's in the Navy, a few jobs for female's in the Artillery, combat pilots, and, at least in the U.S. Army, a few jobs in the Ordinance branch and Chemical Branch can involve combat operations (and said jobs are open to females).
And to be fair, a videogame about being an MP in Iraq would be interesting (and I'd buy it in a heartbeat, but it would be relatively boring as standing in/next to a Humvee for an hour then a small firefight then returning to FOB, rinse, repeat ad-nasum, would be a really boring game.
AND this is all assuming that whatever pretense of "realism" the developer is trying to maintain even cares about any of this to start.