Is there any calendar API for iPad? Please refer to image. This is calendar application in iOS7
Yes there is a calendar and reminders API on iOS (up to iOS 6 that is, being the last public SDK. You won't find any help for iOS 7 outside of Apple sanctioned developer forums until the NDA is raised.). Check out https://developer.apple.com/library/ios/documentation/DataManagement/Conceptual/EventKitProgGuide/Introduction/Introduction.html to find more documentation about Event Kit.
As to present visually the calendar, you'll have to resort to your own solution (or use one of the many excellent widgets like these https://www.cocoacontrols.com/search?utf8=%E2%9C%93&q=Calendar ). Note that none of them exactly look like what you're showing in your screenshot but being open source I'm sure you can find a way to customize them to your needs.