The case is made for both positions, here.
My own personal opinion, as someone who majored in history in college and has taught history is that we ought to consider doing a much better job of educating students about the virtues of the discipline and the breadth of American history as a whole before we focus on anything else. I'm not arguing that African-American history isn't important, but rather that most teachers do such a crappy job of providing the basic facts and contexts of a so-called white-dominated subject that there is a corresponding deficiency in U.S. History literacy for everyone. The student who doesn't know who Frederick Douglass is mostly likely won't know who Stephen Douglass is either.
African-American history finds itself tucked into textbooks and teacher manuals but the same basic mediocrity of pedagogy stays constant. So long as football coaches are allowed to teach history with a sloppy, largely uninformed, overly-simplified, wholly disheartening bone-headed approach, thoroughly boring students and leaving them with an impression that history is a dull topic with no real application to their own lives, then, sad to say, no amount of denoting weeks or months to the study of any minority group is going to make much difference. I didn't enjoy my history classes until I got to college and by then, of course, it's much too late for most people who would sooner take math than history if they even choose to attend a university.
Back to Black History Month---I see the arguments for both sides and I think a more synthesized approach would be best, but that would require many teachers to take workshops and continuing education seminars to learn how to correctly incorporate the history of different racial groups and expertly weave them together. If it were me, then I'd make an effort to not discriminate, since a record of the past includes everyone who lived during a certain time period. It's only descendants who leave things out. Those who lived in those times experienced the broad spectrum. Before we devote time and effort towards Gay History Month or Immigrant History Month, let's aim to not have to isolate each minority, making it seems as though they only time we tip our hat to their place in forming who we are collectively to just thirty days (or less) every year.