Could Jesus have refused to go to the cross?

Jesus prays to the Father in the Garden of Gethsemane, distressing over the impending cup of wrath to come. The weight of bearing the sins of people and alienation from the Father put him such spiritual agony that he asked not once but three times that if there were any other way, for the Father to take this cup from him. Nevertheless, he eventually submits to the will of the Father and chooses to go. As the God-man, _could_ Jesus have refused to go to the cross? If he could have, then wouldn't that mean that he disobeyed the Father and would have sinned which would have been against his very nature? Would that have meant that the triune God would've been broken? If he couldn't have, then he wouldn't have willingly gone to the cross but would have been forced to by God the Father. It wouldn't have a sacrifice out of love but out of obligation. Help me understand what the right doctrine is here regarding Jesus' own free will and nature.