Choose Medicare Act This bill requires the Department of Health and Human Services (HHS) to establish public health insurance plans and addresses health insurance costs. Such public plans must be offered on health insurance exchanges and (1) meet the requirements of a qualified health plan; (2) cover benefits at the gold plan level; and (3) cover reproductive services, including abortions. Additionally, the bill makes a series of changes related to health insurance costs, including requiring employers to refer employees to health care navigators to assist with enrollment, establishing an annual limit on out-of-pocket costs for services under Medicare, providing HHS with the authority to negotiate prices under the Medicare prescription drug benefit program, expanding the premium-assistance credit by benchmarking the credit amount to the second-lowest cost gold plan and increasing the income threshold for eligibility, requiring qualified health plans to reduce cost sharing for low-income plan holders, providing funding for states to provide reinsurance to health insurance issuers and to assist individuals with out-of-pocket costs for plans offered through health insurance exchanges, applying the premium rate-setting requirements to large group health plans, and requiring HHS or states to take corrective actions to address unreasonable premium rates set by insurance issuers.