Hospitals that participate in Medicare and Medicaid must maintain a patient grievance process under the Patient Rights Condition of Participation (42 CFR §482.13), with CMS's expectations for how that process runs spelled out further in its interpretive guidance (State Operations Manual, Appendix A).
The regulation itself does not set a fixed number of days for resolving a grievance. CMS guidance calls for grievances to be resolved within a reasonable time frame and has referenced roughly 7 days as generally appropriate; when a grievance needs longer, the hospital has to tell the complainant it is still working on it and provide a written response by a stated date.
Whatever specific acknowledgment and resolution windows a hospital commits to, drawn from CMS guidance, a state requirement, or its own grievance policy, are deadlines that have to be tracked on every open complaint at once, which is exactly what breaks down in a shared spreadsheet or an inbox full of reminders.