The eminent Muslim scholar and renowned Da`iyah, Sheikh `Abdel Khaliq Hasan Ash-Shareef, states: “Islamic Shari`ah is not the only Shari`ah that obligates stoning to death or rajm on Az-Zani Al-Muhsan (the one who had an illicit relationship while he/she was married before or still married). It was related that a Jew and Jewess had committed zina in the lifetime of the Prophet (peace and blessings be upon him) and because they were of the nobles, some Jews said that we may go to Muhammah to judge them so that he may apply a lesser penalty on them than the penalty we have in our religion. He (the Prophet pbuh) issued his judgment according to the Torah which was stoning them to death.
Also, the hadiths on Rajm Al-Muhsan Az-Zani are mutawatir (having an uninterrupted chain of transmission and a group relate it on the authority of a preceding group until it reaches the Prophet).
So, what happened is that the ruling stated in the verse was not abrogated or cancelled even if the verse was abrogated from recitation. However, another ruling was revealed regarding flogging the unmarried fornicator with 100 lashes. And the ruling of Rajm was still there and it did not need a verse to prove it as it was well proven by the mutawatir hadiths and the deed of the Prophet (peace and blessings be upon him) who stoned Ma`iz and Al-Ghamidiyyah to death when they committed Zina while they had been married. This was witnessed and related by a group of the Companions of the Prophet (peace and blessings be upon him). and there are other incidents on Rajm which was proven by Tawatur.
As for `Umar’s saying, it was related in an ahad (solitary) hadith and you should know that anything related to the Qur’an should be related by Tawatur not ahad. And there is disagreement regarding the interpretation and authentitcity of this saying which is attributed to `Umar (may Allah be pleased with him).