There is no such thing as right wing games, to my knowledge at least and I have been gaming for 35 years. The left were the ones to introduce politics into games as they have done with movies, TV and whatnot. They have polarized everything that used to be just non-partisan entertainment. I guess if you're not pushing left wing talking points in your game, your game becomes right wing by default.
That is wrong. I think the labels of left-wing and right-wing are stupid, as politics has much more shades than black and white. But the notion that games are inherently apolitical is also wrong. As a product of our culture that in turn is shaped by politics they obviously transpire these unconscious political beliefs. If you don't believe it, watch 100 years old movies or read 200 years old books. As our culture has evolved this inherent biases are more visible in old cultural products than in current ones.
Naturally there are also conscious attempts at introducing politics in games. Oliver North was an important figure in the Iran-Contra affair and appeared in Call of Duty. Having such a high profile political figure in your game means you give away the apolitical card. Obviously there are games for everything, including even games for genociding jews. And while I wouldn't put a game like America's Army into a left or right camp, it is an obvious political propaganda game. The list is endless if you want to look at it.
So in conclusion: there exist games that have a political message - and this in every conceivable political flavor. Also games that aren't intended to be political at all might have unconscious political beliefs baked into their cultural background (as all media does). The unconscious part by the way is neither left or right, as it is about something deeply ingrained that is shared by all mainstream-politicians. For instance, these days everyone agrees slavery is bad, so we can assume it is deeply ingrained. But that wasn't always the case (although back then didn't exist games yet).