Can we deal with literary criticism without studying its history?Literary interpretation.
I think that you certainly can use literary theory with a synchronic ("at one moment") perspective. For example, you could read an essay by a particular theorist and apply the language, methods, and concerns of that theorist to a text of your choice. That's actually what most people do.
A fuller understanding of theory, though, probably requires a diachronic ("across time" or "through time") perspective. Important terms such as second-wave feminism or post-modernism don't mean mean if you don't know what came before: Was there a first wave? What does the "post" mean? Ideas do not develop in a vacuum, and theorists and theories are living, changing things. The early Roland Barthes is not the same as the late Roland Barthes.
When I first started really studying literary theory at the graduate level, I enjoyed trying to figure out little geneaologies of theory, to see how one idea or development could lead to another, such as reader response theory leading to some branches of feminist theory leading to lesbian and gay studies and to queer theory. That's not an easy task, though, and I wouldn't require all of my undergraduates to do it.
No matter what lens through which you study a piece of literature, you need to have some context for the lens. For example, you could do a Marxist reading of a text, but if you don't know a lot about Marxism or what inspired Marxism or Marxist literary review, then you will end up doing a rather cursory analysis of the text. The same could be said for a Feminist criticism. You can read a text and pay attention to the female characters, but without an understanding of the role of women in the time period of the text AND what a current feminist would say about gender roles, you wouldn't have a very grounded analysis. If you do a Psychological criticism of the work, you need to understand and apply a psychological theory, not just read for what the main character is thinking.
I do not think that literary criticism, like any other subject matter, cannot be studied without understanding its history. I believe that it is important to have the whole picture when looking at a text or mode of thought. I mean, can one really understand where psychoanalytic analysis comes from without understanding the views from which it came?
Knowing the history, as well as the defining characteristics, seems to come hand in hand. It would be considered superficial if the history was not included in the study.
I think one can actually do quite a good job of literary interpretation but no one will take you seriously unless you can fit your interpretation into the historical frameworks or various "accepted" lenses of criticism. Like anything else, it is sort of a club and they try to keep people out unless they speak the same language and have jumped through the same hoops.