I think the root.zone file would provide what you are looking for. It contains all of the recognized Top Level Domains (.com, .net, .gov, .us, .ru, .cn, etc), along with the name servers IP addresses that service those TLDs.
Keep in mind the root.zone file only gets you one layer deep. As others have pointed out, DNS is a hierarchical, distributed database. There is no central list of every IP address in the world (not feasible).
Say you wanted to go to www.example.com. Your computer needs to know the destination IP address. Using the root.zone file, you would be able to identify the nameservers responsible for all .com domains. Your computer then has to ask the .com nameservers how to resolve example.com. The .com nameservers will help you identify the servers responsible for all example.com sites. Your computer then has to ask the example.com name servers for the IP address of www.example.com.
If you don't trust ICANN much, how much do you trust the registrars that are responsible for maintaining the various TLDs?
Note that in the real world, no one ever uses root.zone directly, and it can go out of date at any time. Instead, there are "root hints" servers that you can query to get this information.
You would not be the first person to have issues with ownership of the root nameservers. Alternate sets of root DNS servers exist. If you are running your own DNS server on your network, you could simply swap out your root hints with this list.
Could you clarify the part on a "list of currently indexed and assigned addresses"? First of all, DNS is about domain names, not (IP) addresses. And are you actually looking for all domains? – None – 2014-04-08T08:01:59.677