Did the Romans ever hold themselves responsible for crucifying Jesus Christ?

I wouldn’t say that the Romans hold themselves as “guilty” or “responsible” for crucifying Jesus Christ. Jesus Christ, to the government of the Roman Empire, was above all, a seditionist. He taught bold religious doctrines, and said things about his own divinity that contradicted Roman tradition and sensibilities. He called himself the “King of the Jews,” a title held by Tiberius Caesar alone (John 19:15). Jesus to the Romans was a mere criminal, an insurrectionist that sought to overthrow the Emperor of Rome in his own interests. The only thing that separated Jesus from any other condemned criminal in the empire, was his notoriety, which superseded the notoriety of John the Baptist. The Romans did not think that Jesus was the son of God, or a divine figure, or a royal figure; they saw him as a trouble-making Jew and they executed him as they would a common thief albeit with applied mockery, e.g. the crown of thorns, the purple robe, the “INRI” marker on his cross etc…

So in this respect, I would say that the Romans saw what they did to Jesus as the right and lawful thing to do considering the political circumstances that were present at that time.