My personal belief is at odds with Catholicsm as I understand it. I don’t believe the Jews were as instrumental in the death of Jesus as many believe. His death took place at a time when thousands of jews had been routinely crucified by the Romans. I find it hard to believe that many jews had the stomach to actively lobby to have one more of their own put to death in that way, let alone to stand around and watch it.
But taking everything in context and assuming the validity of the story…
First, I don’t think Jews would ever use crucifixion as a method of execution. After an execution, in some cases, they might hang a body from a tree or post, but never for a full day/night.
Second, assuming Jesus did everything the Christian Bible says he did and nothing else, I don’t see in there anything that would trigger an execution sentence under Jewish law, except perhaps the use of God’s true name as a blasphemy (which I don’t know if Jesus is said to have done).
He does seem to have violated the Sabbath by gathering food with his disciples (unless they physical well being was in immediate peril).
But I should point out that historically, it was exceedingly rare – and I mean exceedingly rare, for the Jewish court to impose a death penalty on someone, especially for a breach of ritual law (as opposed to punishment for murder, for example). The Talmud teaches that if one person received the death penalty in 7 years, that particular Sanhedrin (Court) was considered exceedingly bloody.