No, Damian Wayne was not adopted. He is the biological son of Bruce Wayne and Talia al Ghul.
Damian Wayne's Biological Parentage
Damian, known by his birth name Damian Al Ghul, is the only biological child of Bruce Wayne and Talia al Ghul. This means his connection to both individuals is through direct genetic lineage, rather than through the legal process of adoption, which integrates a non-biological child into a family.
Circumstances of His Birth
The foundational relationship between Bruce Wayne and Talia al Ghul, which led to Damian's birth, occurred during a period when they were married. This marriage coincided with Bruce's intensive training with the League of Shadows. Following this time, Bruce departed, and he remained unaware of Talia's pregnancy until much later, long after Damian had already been born and raised under his mother's care.