The Arabization of the Fertile Crescent started well before the birth of Christ with the rise of small Arab-ruled states such the Nabataean kingdom and the kingdom of Osroene in Edessa. It wasn't so much that people from the Arabian peninusla were moving into the Fertile Crescent, per se, but rather that the already existing peoples of the Fertile Crescent were becoming part of in an increasingly powerful Arabian cultural and economic network linking the Fertile Crescent, Hijaz, and Yemen. Those who were to become the elites within this burgeoning civilization were known to others as Arabs, regardless of where they originated. The Arabs were the traders and merchants who ferried goods all the way from Oman to Constantinople. They came to be the rulers of the trading posts and oases of the Near East. They became incredibly wealthy and were able to set up small states as a result. They developed a common culture, pantheon, and language that was equal parts Aramaic and South Arabian in character.

Furthermore, by the beginning of 7th century, peoples across the Middle East and North Africa had largely lost their distinctive ancient cultures and languages. There were no Phoenicians and Babylonians and Egyptians. Only Christians and Jews speaking a variety of languages such as Aramaic, Greek and Coptic, in addition to Arabic. After the rise of Islam, these identities and languages continued to exist, and they exist right up until the present day, though almost every Middle Easterner is at least bilingual in Arabic now. It wasn't until the 19th century that the various Arabic-speaking peoples of the Middle East began to view themselves as one 'Arab nation'.

tl;dr - It was a long, drawn out process that started way before Islam and still isn't fully complete to this day.