Community Connect Grant Program Act of 2023 This bill reauthorizes through FY2028 and otherwise modifies the Community Connect Program, which provides grants for high-speed broadband service in rural, economically challenged communities without such service. In particular, the bill establishes statutory minimum broadband speeds that must be provided under the grants. (Currently, minimum speeds are defined by the Federal Communications Commission.) Further, the bill specifies that grant funds may not be used to duplicate a service that will be provided in the future pursuant to an enforceable commitment under another broadband funding program. (Currently, grant funds may not be used to duplicate an existing service.) The bill also makes more areas eligible for grant-funded projects by increasing the broadband service capacity that is considered inadequate.