Unlearning has emerged as a technique to efficiently erase information of deleted records from learned models. We show, however, that the influence created by the original presence of a data point in the training set can still be detected after running certified unlearning algorithms (which can result in its reconstruction by an adversary). Thus, under realistic assumptions about the dynamics of model releases over time and in the presence of adaptive adversaries, we show that unlearning is not equivalent to data deletion and does not guarantee the "right to be forgotten." We then propose a more robust data-deletion guarantee and show that it is necessary to satisfy differential privacy to ensure true data deletion. Under our notion, we propose an accurate, computationally efficient, and secure data-deletion machine learning algorithm in the online setting based on noisy gradient descent algorithm.