I would imagine Romans might have been brought up to feel a sense of shame over this event in a way that might be similar to what Germans must feel over the events of World War 2. Then again, it seems as if early Romans were very proud and not a culture that would criticize state actions, at least publicly. Can anyone elaborate on how early Romans dealt with this? I believe Pontius Pilate himself committed suicide largely over his guilt.
I can't address the first part of the question in depth, but i can tell you that Pontius Pilatus did not commit suicide. Matter of fact, we know very little about him. The only archaeological proof of his life is the [Pilate Stone] (https://en.wikipedia.org/wiki/Pilate_Stone). The only information we gather from this inscribed stone is that he was Prefect of Giudea under emperor Tiberius. This data is in accordance with the information found in the four Gospels. Other than that, we know nothing about his life, before and after his tenure as prefect. We have no knowledge of him committing suicide, and there is no information suggesting he became Christian. It is very unlikely he converted, since for many decades Christian converts were mainly Jews and Christianity didn't become widespread among Romans until at least 50 years after Jesus' death and resurrection.
The Acts of Pilate, contained in the [Gospel of Nicodemus] (https://en.wikipedia.org/wiki/Gospel_of_Nicodemus) are believed to be apocryphal, hence they cannot be used as a reliable history source.
The Roman empire wasn't 'converted' to Christianity overnight. The first emperor to convert to Christianity was Constantine and he was only baptized on his deathbed.
Constantine did not 'convert' the empire though. What he did was more in the line of adopting the organizational structure of the church and bishops, because the Roman bureaucracy was by then a mess of corruption. So, Christians were appointed as officials, but pagan religion was by no means banished, nor was any attempt made, in so far as I know, to justify the crucifixion.
Furthermore, by Constantine's time, there were loads of different faiths that proclaimed themselves to be Christian. The Council of Nicea in 325 AD, ordered by Constantine, was meant to put an end to that. Among other things, it instituted the orthodoxy of the divinity of Christ. What their opinion was about the justification of the crucifixion, I don't know. I'm not sure it was even an issue.
But, again, the whole of Roman culture was not converted to Christianity by that time, so no cultural consensus can have existed about the crucifixion or its justification. The Roman empire did become more and more Christian, as subsequent emperors were all Christian, and like Constantine, appointed Christians as officials, so pagans were more and more left out of the power structure. (A notable exception was emperor Julian the Apostate (361-363), who tried to revert the empire back to paganism, but he died before he could really put his mark on the empire and his successor was Christian again.)
One of the most influential Christian officials in the late Roman empire was Saint Ambrose, but I can't find anything on his views on the crucifixion. Saint Augustine, his pupil, acknowledged that it was a dilemma, but he concluded that it was all part of the divine plan, set in motion at the point of creation. However, it is unclear in how far this was a view that was shared by others.
In short, it is hard to make a statement about Roman culture's view of the crucifixion after the conversion to Christianity, when the conversion itself was so gradual and took such a long time and there was so much debate going on, and the crucifixion wasn't particularly high on the list. The death and resurrection were much more topics of discussion (see the Arian heresy) than the justification of the crucifixion.
Dante, in the Divine Comedy, like Saint Augustine, views the role of Rome as first an worldly empire and then a divine institution as an important instrument in God's plan. In other words: the prosecution by the Romans and the crucifixion had to happen, for the European people to eventually become Christian. But, this was a medieval view, not the view of the Roman empire, which, by that time was (in the West anyway) almost a thousand years gone.
Edit: I would further like to add, that the stories of Pilate's guilt and suicide are stories that spring into existence only centuries later.
The fact is, Pilate was in Roman views a minor official, his death is not officially recorded (as far as we know). This legend forming can be interpreted as signifying a change in view on the crucifixion, sure, but it should not be taken as historical evidence on Pilate or any guilt towards his actions.