I don’t think it’s accurate to say that because Bush had negotiated for an Iraq pullout in 2011, that Obama was bound by that. He could have done whatever he wanted, as he often did. So I think that’s kind of a dubious claim.
Yes, he could have kept our forces there against the will of the Iraqi government simply by re-invading, or knocking over that government and installing yet another government that agreed to our continued presence. That was definitely an option available to him.
The US is not magically released from agreements and treaties it makes with other governments simply because the presidency changes hands. Nobody would bother signing a treaty with us if that were the case.