Uh, convenient conclusion, but remember the bit in the gospel where Jesus stops the crowd from stoning the adulteress? Seems like, whether formally decreed or not, stoning of fallen women was, indeed, a part of the culture.
The adulteress woman that Jesus dealt with, was a set-up in order to trap Jesus. In Jewish law there had to be two witnesses to the crime before stoning was used. The witnesses were grilled intensely to ensure the truth of the crime, before the person was convicted.
The men who brought the woman accused of adultery were most likely voyeuristic buddies of the unapprehended man in the endeavor. It takes two to commit adultery. They said they caught her in the act.
By Jesus possibly revealing the knowledge of who their adulterous buddy was (writing in the dirt for all to see), and the accusers voyeuristic witnessing in order to set up the adultery, would have discouraged the witnesses before they could go through the grilling. They decided to give up on being convicting witnesses.
Josephus tells us that these types of stonings would happen only once every seven years. A rare occurrence considering the propensity for adultery among people.
The Israelite law did, in fact, require stoning of adulterers. The scriptural account you are refering to has two interesting points.
The man was not being chased down - only the woman. The law required that BOTH be stoned. Stoning only her would have been an injustice.
She was being chased down by a mob. We can't be certain, but this appears to have been a group of vigilantes. The law required a system of judges and witnesses. Vigilantism was not in keeping with the law.