Healthcare edge devices (HEDs) can create effective Electronic Health Records (EHRs), thereby enhancing the patient care experience. Nonetheless, EHRs contain a vast amount of private information, necessitating strict access control. Attribute-based encryption (ABE) offers a promising solution for the secure storage and access control of EHRs. However, current implementations encounter several challenges. Firstly, extant schemes are hindered by inefficiency, requiring too many exponentiation calculations and bilinear pairing operations, imposing significant computational demands on HEDs. Secondly, due to the accuracy requirement of medical data and the semi-trustworthiness of the cloud, the verification of medical data decryption becomes imperative. In this paper, we propose an efficient outsourced EHR management scheme using verifiable attribute-based encryption, which ensures data verifiability and enables fine-grained access control. Our scheme significantly reduces the computational overhead for HEDs by outsourcing computation to the CSP, while also supporting a large universe of attributes. Our scheme is proven to be secure and efficient by rigorous security analysis and performance evaluation.