The complexity of building, deploying, and managing cross-organizational enterprise computing services with self-service, security, and quality assurances has been increasing exponentially in the era of hybrid multiclouds. AI-accelerated material discovery capabilities, for example, are desirable for enterprise application users to consume through business API services with assurance of satisfactory nonfunctional properties, e.g., enterprise-compliant self-service management of sharable sensitive data and machine learning capabilities at Internet scale. This paper presents a composable microservices based approach to creating and continuously improving enterprise computing services. Moreover, it elaborates on several key architecture design decisions for Navarch, a composable enterprise microservices fabric that facilitates consuming, managing, and composing enterprise API services. Under service management model of individual administration, every Navarch microservice is a managed composable API service that can be provided by an internal organization, an enterprise partner, or a public service provider. This paper also illustrates a Navarch-enabled systematic and efficient approach to transforming an AI-accelerated material discovery tool into secure, scalable, and composable enterprise microservices. Performance of the microservices can be continuously improved by exploiting advanced heterogeneous microservice hosting infrastructures. Factual comparative performance analyses are provided before the paper concludes with future work.