No – and the confusion on which the charge rests is instructive enough to be taken apart.
What Hegel does not claim
He does not claim that one may say A and not-A at will. Anyone who did could no longer assert anything determinate, and Hegel’s whole procedure consists in setting up very determinate claims in a determinate order. A thinking to which everything were equally valid is for him expressly the worst – it is the scepticism that „always sees in the result only pure nothing“.
What he does claim
That things, and the concepts with which we grasp them, turn into their opposite at their own limit, and that it is precisely from this that movement arises. Contradiction is in him not a property of sentences but of states of affairs:
… if there were any question of ranking, and the two determinations had to be held apart as separate, then contradiction would have to be taken as the deeper and more essential. For identity, as against it, is only the determination of the simple immediate, of dead being; but contradiction is the root of all movement and vitality; only in so far as something has a contradiction within itself does it move, have drive and activity.
Science of Logic II, „Contradiction“
Lawyers have a better ear for this than logicians. A contract is for Hegel expressly a contradiction (§ 72): I remain owner in ceasing to be one. No lawyer takes that for a mistake in thinking; it is the description of what happens in a transfer of title. Likewise wrong (§ 82): it is the negation of right which does not remove right but compels its restoration.
Why this is not an abolition of logic
What is decisive is Hegel’s concept of determinate negation. A negation that only says „not“ ends in empty nothing and carries nothing. A determinate negation is the negation of this matter and retains its content: „so that the result essentially contains that from which it results – which is properly a tautology, for otherwise it would be an immediate and not a result.“ It is precisely this that keeps every step checkable.
Scholarship now reads it this way: Hegel does not deny the principle of contradiction, he reinterprets the formal principles from a standpoint at which they show their limit. The charge belongs among the legends that Jon Stewart records separately.
Sources. Jon Stewart (ed.), The Hegel Myths and Legends, 1996, there „The Myth that Hegel Denied the Law of Contradiction“ with contributions by Robert Pippin and Robert Hanna · Science of Logic II, Second Section, „Contradiction“. – In the edition: § 72, § 82, § 31. – In the glossary: Contradiction, Determinate Negation, Dialectic.