I have read the speech. Many, many times. It most certainly doesn't mean the opposite of that. He states it even more clearly earlier in the speech:

"The marvelous new militancy which has engulfed the Negro community must not lead us to a distrust of all white people, for many of our white brothers, as evidenced by their presence here today, have come to realize that their destiny is tied up with our destiny."

He does talk about the bad check that America has given to black people. He's right. At no point does he refer to it as a white check. I'm not sure where you got any other idea.