Social network analysis (SNA), or the application of network analysis techniques to social media data, is an increasingly prominent approach used in computational public health research. We conducted a systematic review to investigate trends around SNA applied to social media data for public health and epidemiology while outlining existing ethical practices. Following PRISMA guidelines, we reviewed articles from Web of Science and PubMed published between January 2019 and February 2024, leading to a total of 51 papers surveyed. The majority of analyzed research (69%) involved studying Twitter/X, followed by Sina Weibo (16%). The most prominent topics in this timeframe were related to COVID-19, while other papers explored public health topics such as citizen science, public emergencies, behavior change, and various medical conditions. We surveyed the methodological approaches and network characteristics commonly employed in public health SNA studies, finding that most studies applied only basic network metrics and algorithms such as layout, community detection, and standard centrality measures. We highlight the ethical concerns related to the use of social media data, such as privacy and consent, underscoring the potential of integrating ethical SNA with more inclusive, human-centered practices to enhance the effectiveness and community buy-in of emerging computational public health efforts.