This is a case of Randomized Rumor Spreading. Normally, one uses one of the following three models for rumor spreading in graphs:
Push model: In each round, every white vertex calls an adjacent vertex, and colors them white.
Pull model: In each round, every black vertex calls an adjacent vertex, and if they call a white vertex, they become white themselves.
Push/Pull model: In each round, both of the above happen.
When you consider different random graph models, such as the Erdős–Rényi Model, or the Preferential Attachment model, you can get bounds of $\log n$ or better. This also holds for complete graphs.
Actually, there would be cases for your model where the vertices do not become white. Consider for example a path on 4 vertices, where the end vertex is white. In each round, you pick a perfect matching, and there is only one of those. In this case, the other end of the path will never become white.