Imam al-Shafi'i (may Allah have mercy on him) said: "Because the Messenger of Allah (peace and blessings be upon him) permitted some people in Makkah to remain there after they accepted Islam, among them Sayyiduna al-Abbas bin Abd al-Muttalib (may Allah be pleased with him) and others, since they did not fear trial (regarding their religion)."
Urwah bin al-Zubayr (may Allah have mercy on him) says that Sayyiduna al-Abbas (may Allah be pleased with him) accepted Islam, continued in his office of giving water to the pilgrims in Makkah, and did not emigrate.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب السير / 17762