Not much more than a major investigation.
Cable disruptions happen. When they do, traffic is rerouted. Usually you have plenty of alternative cables to reroute to along the same path. There are 15 cables between the United States and Europe, although there are other Atlantic cables between other countries and continents. If these were to be cut, we would reroute data through the pacific cables. This would cause increased stress for sure, and might result in some increased ping time to Europe, but communication will continue.
There would certainly be an investigation. Multiple cables being cut at the same time would either be assumed to be a natural event (earthquake) or possibly the Russians. This would raise alarm at the Pentagon, but seeing as how it’s only you cutting the cables, the diplomatic tension would be temporary.
This NPR article mulls over the same question (due to concerns about Russian submarine interference).
Even if all Atlantic and Pacific cables were cut by a James Bond villain, satellites will still provide for cross-continent communication to take place. They won’t be capable of carrying the entire load of public internet traffic, but critical communication will continue. The cables are also only facilitating connections to foreign servers. In the U.S., your average user probably wouldn’t even notice the cut given the high number of services based in the U.S. Worldwide there might be more issues, but most major U.S. based internet companies have servers on other continents.