We say: I don't know whether he is right.
Is it acceptable to say: I don't know he's right or wrong.
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We use "if" or "whether" to introduce clauses after verbs of doubting:
I don’t know if I can drive. My foot really hurts.
Look at If or whether: indirect questions at the Cambridge Dictionary.
I don't know whether he is right or not
I don't know whether he is right or wrong