While there is a reasonable doubt about that scripture being valid, as it does not appear in the oldest manuscripts available-- where there is a reasonable doubt, do we convict by it?
Nevertheless, if we assume that it is valid, there are a lot of possibilities about the unknown variables. Concerning the man she was caught in adultery with-- we don't know if he got away unidentified, if she was caught in his home/tent/whatever and he wasn't there at the time-- which would raise the question Jewish legal proceedings of 2 or 3 witnesses actually seeing them in the act. But I have long thought that best of all among the possibilities is that the whole things was a framing. And maybe what Jesus wrote on the ground had to do with that; he wrote the names of the accusers there who had all committed adultery with that particular woman. It's hard to see why else they would just turn and leave from the whole incident. Anyway, they had wanted to get rid of her, as well as to get rid of Jesus, and they saw the 2-birds-with-one-stone setup and went for it. The woman herself, knowing if she were not condemned for the particular case there were others right there, so it was no use denying anything (that is, if she could/would speak up at all, which is questionable).
Conclusions to be drawn if we assume the validity of the passage:
Did Jesus abolish the death penalty in all cases? No; he did not make such a blanket statement.
Did he abolish it for adultery? Maybe.
Did he abolish it for anything short of murder? We don't know.
Did he abolish the Noahic Covenant (which includes putting a murderer to death)? I certainly don't think so-- its validity, at least in part, is confirmed in Acts 15 when the apostles said gentile converts must not eat blood.
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When Jesus made the point that only the one without sin may throw the first stone, what was that about? If He had thrown the first stone, being the one allowed by his statement, could the others then have thrown a bunch and everything then rosy? Did he actually turn his back on the Mosaic Law, destroying it after all? Is a Christian not allowed to sit on a jury, let alone become a judge, and assign guilt or not to someone accused? Can even parents not pronounce sentence upon their children when they've done something wrong, as they are not without sin themselves?
Or--- is all this about Jesus and his role among mankind which is not to condemn, but to save? He didn't heal every diseased person or right every wrong-- at times he healed, at times he set people straight. But he often refused to get involved, such as when the man asked him to tell his brother to share their father's estate. God is salvation was both his name and his function. Why would he add to the already-overbalanced opposite position?