If I understand you correctly, you are saying that the act of inviting a woman to a pillow fight is sexism regardless of the gender of the requester. If so then we can agree that the act is equally <what ever label you use>. We may disagree on what that label is. If you label it sexism, that is your choice. I would call that rude or aggressively forward.
However, if you are thinking that it is only sexism if the requester is a male, then I think we'll have to agree to disagree completely because I am not going to buy into such gender based biases.
I'm not sure you do understand. The point is rather, "yes, it is rude and aggressively forward. Added onto that, the fact that someone felt free to do it, is sexism. And, the global environment that it creates is also sexism."
It is a sexist request in the sense that it was constructed by, and constructs, sexism. It does not directly oppress anyone, that is true -- but that is irrelevant, because that isn't the only mechanism of sexism (and isn't even the dominant mechanism).
As long as we are agreeing that it is <what ever> regardless of who does it, I really don't care what you label it. That was my point. You can believe it is sexism. I will believe it is not. I don't have to believe it is sexism to think the behavior is bad. It is bad regardless. However, I believe labeling it as sexism is overly harsh due to what that label implies. I don't believe that label is deserved.
However, if you are thinking that it is only sexism if the requester is a male, then I think we'll have to agree to disagree completely because I am not going to buy into such gender based biases.