My understanding is that he cannot object to the JUDGMENT of Abitrator Goodell, but he can set aside the verdict for bias, or procedural unfairness.
Such as Goodell rejecting case history that he was the losing party in, Goodell changing the reason for punishment, witness not being allowed to testify, and the whole conflict of interest and the bias and unfairness it created.
Once Brady gets the award vacated it will return to a new arbitrator (unless the NFL just gives up) and that arbitrator will make new judgement on the evidence.