This is actually a more complex question than it might appear on the surface. The first issue is whether the concept of "gay persons" existed in the Renaissance. Michel Foucault has argued that the notion of "gay persons" is really a nineteenth century concept, and that in earlier cultures, acts could be defined as gay (e.g. sodomy), but not persons. Thus, it wouldn't really be possible to argue that Hamlet was a gay person because that would not have been an available conceptual category.
As to whether Hamlet would fit under a 21st century notion of gayness, there is no immediately obvious evidence in the play that this would be the case. First, he appears to love Ophelia. Second, his close male friendships seem fairly typical for the period and there is no overt statement that they include sexual activity.