So will I be able to access a multicast group over the internet or
will it only be supported through private networks like in ipv4?
I must correct a assumption you seem to have made here.
If all the routers between you and your destination support it, then Multicast can certainly work over the IPv4 Internet. It is simply blocked or not configured in many places. I suspect this is because multicast is not well understood, and many people believe they do not need it. So they simply do not permit it through their firewalls/routers.
IPv6 certainly is capable just like IPv4 of having Multicast work globally. Only time will tell us if people actually permit multicast through their networks.