Q: I am having problem sharing my calendar publicily.

You are correct that they do not need to be an iCloud user to share your calendar publically.  When they click the link in the email the calendar should have been automatically added to the calendar program on their computer as a subscribed calendar.  If it wasn't, you can send them the link to manually add it.  To do this, go to icloud.com, click on the calendar you want to share, click the sharing icon to the right as you did before and check Public Caldar (uless it's already checked), then copy the entire url listed that begins with "webcal://.  The link includes everything before the "Email link" below it, so be sure to copy the entire thing.  You can then paste this into an email and send it to your friend.  They will then need to find the instructions for how to manuallynadd a subscribed calendar to the calendar program they are using on their computer using the link you sent.