Website Business Appointment Calendar Integration

I need a clean, embeddable appointment calendar for my websites that handles business-project bookings end-to-end. Visitors should be able to view available slots in real-time and confirm an appointment through a straightforward online booking flow — no separate emails or phone calls required.

The calendar must:
• Display and manage business-project slots I set in the back-end.
• Offer an instant online booking form that updates availability as soon as a slot is taken.
• Sync seamlessly with my existing appointment-booking system so confirmed meetings appear on both sides without manual duplication.

If you can also leave the door open for future connections to Google Calendar, Microsoft Outlook, or Apple Calendar, mention how you would structure the code to make that extension simple.

Deliverable is a fully tested calendar module (HTML) integrated into my current site, with clear setup instructions so I can adjust time blocks and blackout dates on my own. The deliverable must be able to be added to "n" number of websites.