Foundational Models (FMs) are emerging as the cornerstone of the biomedical AI ecosystem due to their ability to represent and contextualize multimodal biomedical data. These capabilities allow FMs to be adapted for various tasks, including biomedical reasoning, hypothesis generation, and clinical decision-making. This review paper examines the foundational components of an ethical and trustworthy AI (ETAI) biomedical ecosystem centered on FMs, highlighting key challenges and solutions. The ETAI biomedical ecosystem is defined by seven key components which collectively integrate FMs into clinical settings: Data Lifecycle Management, Data Processing, Model Development, Model Evaluation, Clinical Translation, AI Governance and Regulation, and Stakeholder Engagement. While the potential of biomedical AI is immense, it requires heightened ethical vigilance and responsibility. For instance, biases can arise from data, algorithms, and user interactions, necessitating techniques to assess and mitigate bias prior to, during, and after model development. Moreover, interpretability, explainability, and accountability are key to ensuring the trustworthiness of AI systems, while workflow transparency in training, testing, and evaluation is crucial for reproducibility. Safeguarding patient privacy and security involves addressing challenges in data access, cloud data privacy, patient re-identification, membership inference attacks, and data memorization. Additionally, AI governance and regulation are essential for ethical AI use in biomedicine, guided by global standards. Furthermore, stakeholder engagement is essential at every stage of the AI pipeline and lifecycle for clinical translation. By adhering to these principles, we can harness the transformative potential of AI and develop an ETAI ecosystem.