Zero-Touch Networks (ZTNs) represent a state-of-the-art paradigm shift towards fully automated and intelligent network management, enabling the automation and intelligence required to manage the complexity, scale, and dynamic nature of next-generation (6G) networks. ZTNs leverage Artificial Intelligence (AI) and Machine Learning (ML) to enhance operational efficiency, support intelligent decision-making, and ensure effective resource allocation. However, the implementation of ZTNs is subject to security challenges that need to be resolved to achieve their full potential. In particular, two critical challenges arise: the need for human expertise in developing AI/ML-based security mechanisms, and the threat of adversarial attacks targeting AI/ML models. In this survey paper, we provide a comprehensive review of current security issues in ZTNs, emphasizing the need for advanced AI/ML-based security mechanisms that require minimal human intervention and protect AI/ML models themselves. Furthermore, we explore the potential of Automated ML (AutoML) technologies in developing robust security solutions for ZTNs. Through case studies, we illustrate practical approaches to securing ZTNs against both conventional and AI/ML-specific threats, including the development of autonomous intrusion detection systems and strategies to combat Adversarial ML (AML) attacks. The paper concludes with a discussion of the future research directions for the development of ZTN security approaches.