Toxicity mitigation consists in rephrasing text in order to remove offensive or harmful meaning. Neural natural language processing (NLP) models have been widely used to target and mitigate textual toxicity. However, existing methods fail to detoxify text while preserving the initial non-toxic meaning at the same time. In this work, we propose to apply counterfactual generation methods from the eXplainable AI (XAI) field to target and mitigate textual toxicity. In particular, we perform text detoxification by applying local feature importance and counterfactual generation methods to a toxicity classifier distinguishing between toxic and non-toxic texts. We carry out text detoxification through counterfactual generation on three datasets and compare our approach to three competitors. Automatic and human evaluations show that recently developed NLP counterfactual generators can mitigate toxicity accurately while better preserving the meaning of the initial text as compared to classical detoxification methods. Finally, we take a step back from using automated detoxification tools, and discuss how to manage the polysemous nature of toxicity and the risk of malicious use of detoxification tools. This work is the first to bridge the gap between counterfactual generation and text detoxification and paves the way towards more practical application of XAI methods.