I'm asking about whether he is married or not.
I'm asking whether he is married or not.
Are they both correct?
I'm asking about whether he is married or not.
Technically this is not asking about marriage status. It concerns you stating what the topic of conversation is. In that sense, it is grammatically correct. Think of it as, "I'm asking about whether he is married, not whether he loves her."
However, both your sentences are things native speakers would say and expect to mean the same thing. The words "about whether" are sort of awkward in that construct. It is like someone merged the two sentences:
I'm asking about his marriage status.
I'm asking whether he is married or not.
So, yes, you can and should drop the about, unless you actually mean it in the technical sense.