There is no evidence for such a claim. Paul states that he had authority from the High Priest to persecute (Acts 22:5). He does not appeal to any authority from Caesar. Besides, Paul was a Pharisee (Phil 3:5) -- members of this party kept clear of the Romans, lest they become ceremonially impure (John 19:28).
I heard from a preacher that when Paul was watching Stephen being killed (Acts 7:58), and as he continued to persecute Christians, he was doing what he had been trained to do by the Romans. Is this true? - R.L.
Technically, Paul could no more execute a Christian than the High Priest could have executed Jesus. Yet extrajudicial killings happened all the same. No one needed "training" in order to kill somebody else. Further, the preacher is evidently unaware of early Christian and Roman history. The first state-sponsored persecution took place in the summer of 64 AD, and another around the year 95, but the Romans hadn't gotten Christianity yet -- not until the 2nd century.