(While the OP might know much of the following, I’ll still give some basic info.)
There are two main high level controls:
Assignment of IP address, those 4 octet numbers like 10.15.1.127. Other than running out of IP addresses (of which the US initially got a quite large fraction.) Not too much of a dispute.
Domain Name Service (DNS). Which maps foo.bar.com to 10.15.1.127. The assignment of the “foo” part is solely up to the company that owns the bar.com domain. Not too many people have a problem with the assignment of the “bar” part. It’s the control of the “.com” part that is the biggest issue. Such suffixes are called “TLDs” for “Top Level Domains.” The control of various TLDs has been given or sold to various organizations. “.com” is controlled by a US company Verisign that just ticks people off.
A lot of non-US businesses want to have “.com” domains without being beholden to Verisign (and in turn, from their point of view, the US Govt). “BigBucks.co.uk” doesn’t get as much traffic as “BigBucks.com”
Note also, that the overwhelming majority of folks overseas currently think of the US Govt. as a bunch of arrogant, ignorant ,out-of-control cowboys. If the US decided one day that France wasn’t subservient enough, the US government could take down “.fr” and any French domains they wanted. That would make it very hard to get to those sites.
Given the climate in Washington DC, this is viewed overseas as a distinct possibility.
Solution: take control of DNS/TLDs away from the US Govt. Put it in the hands of a UN approved organization.
New problem: The UN is not viewed at all well by the folks in charge in DC.
If no resolution is forthcoming, Bad Things could happen. E.g., there could be 2 Internets. The US one and the rest of the world. Still physically connected, but the easy WWW url system we are so used to breaks down.
What if amazon.com in the US goes to an online book store, while overseas it goes to a Brazilian tourist site?
If this were to really happen, people would come up with workarounds but they wouldn’t be easy. E.g., an extra suffix like “.them” would be added for users in the US to access overseas sites. But the web master’s overseas might have hard links within their pages that of course don’t have “.them” embedded. More software workarounds for that. Then more patches. Etc.
So many people, all over the place, would get royally ticked off. Once people get ticked off to that degree, peaceful resolutions are less likely.