Praise be to Allah.
Passing the Black Stone and acknowledging it, after praying two rak’ahs of Tawaaf and drinking from the well of Zamzam and before going out to perform al-Saa’iee is proven to be part of the saheeh Sunnah, but it is mustahabb (encouraged), not waajib (obligatory), so there is nothing wrong with not doing it.
The proof (daleel) that it is in accordance with the Sharee’ah is the hadeeth of Jaabir (may Allaah be pleased with him), reported by Muslim, in which he described the Hajj of the Prophet :
". . . Then he went back to the Rukn (the corner of the Ka’bah where the Black Stone is situated) and acknowledged it, then he went out through the gate towards al-Safaa’."
Al-Tirmidhi reported from Jaabir that: "When the Prophet came to Makkah, he entered the Mosque and acknowledged the Black Stone, then he passed to the right of it and ran for three (circuits of Tawaaf) and walked for four. Then he came to al-Maqaam and said: ‘Take Maqaam Ibraaheem as a place of prayer.’ He prayed two rak’ahs with the Maqaam between him and the House (Ka’bah).’ Then he came to the (Black Stone) after praying two rak’ahs and acknowledged it. Then he went out to Safaa’ and I think he said, ‘Safaa’ and Marwa are of the Symbols of Allaah [al-Baqarah 2:158].’" (Sunan al-Tirmidhi, 784).
Al-Tirmidhi said: The hadeeth of Jaabir is a saheeh hasan hadeeth and the scholars ruled that we should act in accordance with it).
Al-Nawawi, may Allaah have mercy on him, said: "This is evidence in favour of the opinion of al-Shaafi’i and other scholars, that after performing Tawaaf al-Qudoom (the Tawaaf of Arrival) and praying behind the Maqaam, a person should go back to the Black Stone and acknowledge it, then go out through the Gate of Safaa to perform al-Saa’iee. The scholars agree that this acknowledgement is not obligatory, but it is Sunnah; the one who does not do it is not obliged to offer a sacrifice in compensation. (Saheeh Muslim bi Sharh al-Nawawi, 8/428).
And Allaah knows best.