Wireless medical sensor networks (WMSNs) play a significant role in increasing the availability of remote healthcare systems. The vital and physiological data of the patient can be collected using the WMSN via sensor nodes that are placed on his/her body and then transmitted remotely to a healthcare professional for proper diagnosis. The protection of the patient's privacy and their data from unauthorized access is a major concern in such systems. Therefore, an authentication scheme with a high level of security is one of the most effective mechanisms by which to address these security concerns. Many authentication schemes for remote patient monitoring have been proposed recently. However, the majority of these schemes are extremely vulnerable to attacks and are unsuitable for practical use. This paper proposes a secure three-factor authentication scheme for a patient-monitoring healthcare system that operates remotely using a WMSN. The proposed authentication scheme is formally verified using the Burrows, Abadi and Needham's (BAN) logic model and an automatic cryptographic protocol verifier (ProVerif) tool. We show that our authentication scheme can prevent relevant types of security breaches in a practical context according to the discussed possible attack scenarios. Comparisons of the security and performance are carried out with recently proposed authentication schemes. The results of the analysis show that the proposed authentication scheme is secure and practical for use, with reasonable storage space, computation, and communication efficiency.