Depending on how simplistic of a view you want to take on the topic, I'd say that it is not correct. While the US has enormous influence and vast-ranging powers, it does not have complete, unilateral control the internet.
For example, .to, .ch, .se and .is are only subject to Tongan, Swiss, Swedish, and Icelandic law respectively, which is why they are popular alternatives to those wary of for example .com. US authorities cannot simply take down a .se domain, but it could probably leverage political pressure to try to convince Sweden to take action. This is less likely to succeed in countries hostile or indifferent to the US.
However, many ccTLDs (country code top level domains) are under US control by being administered by or through US universities. The University of Puerto Rico administers several Caribbean and American ccTLDs.
The US does not control the ISPs, physical cables, data centers, POPs, and other parts of the internet that's outside of US jurisdiction. But ICANN and IANA are very important organisations for the internet's architecture and both are US companies. Since a 2016, this influence has been decreasing through an internationalisation process.
