Health misinformation, most visibly following the COVID-19 infodemic, is an urgent threat that hinders the success of public health policies. It likely contributed, and will continue to contribute, to avoidable deaths. Policymakers around the world are being pushed to tackle this problem. Legislative acts have been rolled out or announced in many countries and at the European Union level. The goal of this paper is not to review particular legislative initiatives, or to assess the impact and efficacy of measures implemented by digital intermediaries, but to reflect on the high constitutional and ethical stakes involved in tackling health misinformation through speech regulation. Our findings suggest that solutions focused on regulating speech are likely to encounter significant constraints, as policymakers grasp with the limitations imposed by freedom of expression and ethical considerations. Solutions focused on empowering individuals – such as media literacy initiatives, fact-checking or credibility labels – are one way to avoid such hurdles.