There have been incredible recent advances in the research on how genes and experience influence development. One of the most important things to take away from the newer research is that it's not a question of two independent forces (nature and nurture) exerting influence in different proportions --- your genetic make up affects the way experience affects you, and your experience has the potential to change the genetic component itself. Yes, nurture affects nature.
We learn about the genetics of eye color or blood type in our high school biology classes, and then we're left to assume that there are similar processes in place for the rest of our genetic inheritance --- unfortunately, it really doesn't work that way. Once you get past the most basic traits, the genetic part is actually very difficult to understand, in part because its expression often depends on environmental circumstance. Even something like height, which seems like it should be pretty straightforward, is surprisingly complex (if you read the linked article, don't stop after the first paragraph; it opens with a simple assertion, but the rest of the article explains all of the main exceptions and caveats to that pattern). In fact, even eye color is more complex genetically than you were probably taught. If you start talking about cognitive and behavioral traits, like intelligence, linguistic ability, or cooperation, there is literally no trait that has a clear, simple genetic cause. For example, IQ has been rigorously studied for decades, including a tremendous amount of research on its genetic component. We know from this research genetics definitely plays a part in IQ, but no one has been able to identify the individual genes that control it. (Some relevant reading, if you want to learn more: article on genetics of race and IQ; summary of evidence on genes and IQ).
All of this is to say you can't really "pass on" cooperativeness to your offspring the way you might pass on your blood type. To address your question directly,
She believes that it is something he can learn through playing with others, while I believe it something that is passed on based on my stubbornness.
Your child will have roughly 50% of his genetic material from you. Any complex trait like cooperativeness will be influenced by a huge number of individual genes, each contributing only a tiny bit to the behavior. Your child will likely have some but not all of your constellation of genes that contributes to your behavior as cooperative or not --- since genes work together in complex ways, you have no idea whether the pieces of cooperativeness genetics your child will inherit will work together the same way yours do. This is all before we even consider the fact that cooperativeness might be influenced by experience/environment. As it turns out, environment is probably more important than genetics in the development of cooperative behavior: I was able to find one study on this, and they found that only 38% of cooperativeness appears to have a genetic link. So put together the fact that most of cooperativeness is predicted by environment rather than genes and that the genetic component is likely very complex, and you can see how tenuous your claim that you can "pass on" cooperativeness is.
To sum up...
Complex behaviors (and, actually, even many simple traits) can't adequately be described as "inherited or learned", or even as "inheritance plus experience" --- your genetic inheritance is complex, flexible, and it can change over time depending on your environment.
Any cognitive or behavioral trait has complex genetic determinants. There's no one gene, or small set of genes, that will determine whether you do or don't show some behavior (with the exception of inherited forms of psychopathology, which is outside the scope of this question).
The little we know about the inheritability of cooperativeness suggests that genes play a relatively small role (compared to something more heritable like IQ, roughly 70% of which may be influenced by genes).
Although you're right that your genes might make a difference in some subtle ways to your child's cooperativeness, on the whole your wife wins this argument. :)