You can't prove that from that. The crucifixion of Christ, from the earthly point of view, was a lynch job pure and simple, and Pilate as one who must answer to God as does all earthly government, had NO legitimate business ordering it. The "authority" in this context is the same as the "authority" given the devil... i.e. permission.
Oh, I agree. Of course. However, the fact that a God-given authority can be abused does not mean that the authority has not been given. There is an old saying in theology: Abusus non tollit sed confirmat substantiam. "The abuse does not destroy but confirms the substance." A rogue cop who wrongly uses deadly force, a deranged soldier who commits a massacre--they are abusing their God-given authority. Even so, that does not do away with the proper use of that authority to wield the sword--whether by policemen or soldiers or a Roman governor or an American criminal justice system.
And I agree that the one passage I cited, John 19:10-11, does not speak directly to the issue of capital punishment. But Jesus there does indirectly acknowledge that the state does have the God-given authority to put to death.
The other passage I cited, Romans 13:3-4, speaks more directly to the God-given authority of government to carry out capital punishment. It is the locus classicus on the subject.