As a male, I have no problem playing games featuring male or female protagonists. However it is impossible not to notice the dearth of female protagonists in video games.
The new Tomb Raider is a step in the right direction, showing a strong competent female who is more than T&A (unlike most of the original Tomb Raider games. And please note I have not played the new one yet, but this is how I have perceived the game from reviews and previews). I enjoyed Heavenly Sword, but the story was lacking a bit. I enjoy all the Metroid games, although I have not yet played Other M. I could never get into Mirror's Edge after playing the demo. Going back to the beginning of PS2, Oni featured a pretty good female protagonist. I should really go back and try to finish that game.
Sadly, these are the only games I can think of with singular female protagonists that seem to offer more than just T&A, though admittedly I have not played every game out there. I am looking forward to Remember Me after seeing the demo a few months back, and hope
There are some games that feature multiple characters, notably the Final Fantasy series (or nearly any JRPG really), where the numbers of each gender seem pretty balanced. Heavy Rain had the female reporter, who I felt was one of the strongest characters in the game. The Tenchu series was pretty good, having both a male and female character to choose from, though the characterization wasn't as fleshed out as it could have been.
When it comes to games like Fallout or Oblivion, I typically create a male character, as this is the character I am most likely going to project myself into. In the above games, the characters (usually) already have a personality all their own, and I cannot affect it. In these kind of games though, it's mostly first person silent protagonists, so I can pretend it is me wandering the wasteland or exterminating demons. However, Dragon Age: Origins had so much character creation choices and interaction between you and the NPCs I ended up making multiple characters of both genders to see how things change from one to the other.
Then there's the female protagonists that just don't work for me. The earlier Tomb Raider games, for instance. I played the first one, and it was okay, but the over-sexualization of Lara in the following games just turned me off. Never played Bayonetta because of how it looked. Played a little of Bloodrayne and though the gameplay was okay, didn't care for the character. Don't care about Dead or Alive, or any game where the work put into the breast jiggle physics outweighs the work put into everything else. Wow, that's actually all I can think of with female protagonists. If we look at side characters though, there's too many females who wear what I can only assume to be dental floss (the woman in Ninja Gaiden, and some from the Devil May Cry series come to mind).
TL

R - Being a male, I am comfortable playing as characters of either gender, though I recognize there are not many female protagonists in video games at this point, but hope things change for the better soon. The new Tomb Raider seems to be a step in the right direction.