There were no major engagements between American and German or Italian ships in the Mediterranean, Arctic or Atlantic. There were two reasons for this: there were comparatively few modern USN ships in the European theatre to be assigned to combat operations, and the German and Italian navies, low on fuel and confidence, rarely sought battle.

The USN sent most of its modern vessels to the Pacific theatre. Comparatively few ships operated in the Atlantic. The carrier Ranger stayed in the European theatre throughout, as it was felt she was unsuitable for the demands of the Pacific, while Wasp operated in the theatre as a way to build crew experience and due to a lack of British carriers. Similarly, the modern battleships Massachusetts and Washington operated in the theatre during the earlier parts of their naval career. The USN's choice to de-emphasise the European theatre reflected two main things: the primacy of naval operations in the Pacific theatre, and the strength of the Royal Navy within the European theatre. The Pacific War was primarily a naval one, and placed much higher demands on the USN's strength. In the European theatre, meanwhile, the Royal Navy was strong enough to overmatch the Kriegsmarine and Regia Marina. It had its own established bases in Europe, and its key bases in the Pacific had been overrun in 1941. In comparison, the USN retained its key Pacific bases, but had no infrastructure in Europe. As such, it made sense for the RN to focus on the European theatre, while the USN focused on the Pacific.

The Kriegsmarine and Regia Marina were both rather unwilling to fight for much of the war. They had limited supplies of fuel, which tended to keep their ships confined to port. They had comparatively few ships, and so were unwilling to risk them in combat. This greatly reduced the number of opportunities for American ships to engage in combat. As an example, the USN battleship Washington and cruisers Tuscaloosa and Wichita formed part of the covering force for the Arctic convoy PQ17. This convoy was threatened by the German Operation Rösselsprung, which was a planned attack on the convoy by the battleship Tirpitz, and cruisers Lutzow, Admiral Scheer and Admiral Hipper. However, thanks to Allied sightings of the force, the convoy scattering, and Lutzow running aground*, the Kriegsmarine worried that they were going to sail into a trap for which they were not prepared, and withdrew without a surface attack on the convoy.

There was, however, an engagement between the USN and the French Navy in 1942. During Operation Torch, the Allied landings in French North Africa, a strong USN force approached the port of Casablanca, to begin landing troops around the port. Casablanca was one of the more significant ports for the remaining ships of the Vichy French Navy. The force protecting the port included the incomplete battleship Jean Bart, light cruiser Primaguet and nine destroyers. Over the course of the 8th November, most of these forces attempted to sortie from the port to engage the American landing force. Primaguet and five of the destroyers were sunk, while Jean Bart was damaged by Massachusetts. On the 10th, Jean Bart would again attempt to engage, and would be sunk by dive bombers from Ranger, though she would later be salvaged.

As /u/thefourthmaninaboat notes, there really were not any major surface encounters in the proper sense of the word. Most of the USN's surface encounters with the European Axis tended to be very small-scale compared to the Pacific battles and often entailed units smaller than destroyers after the North African campaign. That said, there were some battles that did take place.

A good many of these actions involved German torpedo boats (S-boats, also confusingly termed E-boats in some Anglophone literature and also not to be confused with actual torpedo boats, which tended to be larger than the PT-boat-sized S-boats) or other small craft intercepting Allied naval traffic. The invasion of Southern France did witness two small battles between USN surface units and German ships. The American destroyer Somers on 15 August 1944 engaged with two ships the Germans had captured UJ6081 (the former Italian corvette Camoscio ) and SG21 (the French aviso Amiral Sénès and sank them both. An American surface patrol engaged the Kriegsmarine two days later off of La Ciotat with the destroyer Endicott and two RN gunboats sinking UJ6073 and UJ6082. There was a more serious encounter off of Imperia in the Ligurian Sea on 2 October 1944 in which the USN destroyer Gleaves tangled with three torpedo boats engaged in offensive mining operations. This night action though was confusing and neither side understood the other's intents; the Gleaves captain initially thought fire from the German ships was from shore batteries and the Germans' commander overestimated the size of the Allied enemy they were facing.

Not all of the American surface actions in European waters were responses to German incursions. There USN was involved in offensive patrols to break up or destroy German sea traffic in France. While RN ships outnumbered USN ones, the Americans were present in these offensive sweeps and interceptions. The Kriegsmarine's 24th Minesweeper flotilla operated in the waters of the Gulf of St. Malo and the German-occupied Channel Islands. The 24th's principle American opponents were PT Squadron 30 and PT Squadron 34 which operated out of recently-captured Cherbourg. These PT boats operated with larger destroyers and destroyer escorts, such as on the night of 6-7 August 1944 when the DE Maloy and PT503, PT500, PT507, PT508, and PT509 tried to intercept German minesweepers off of Jersey. The American attack was not successful and the task force lost PT509. Six days later the DE Borum and PT500 and PT502 engaged with the Germans again with indecisive results. Borum and her PTs took part in a mixed RN force outside of Guernsey on 13-14 August against a small force of four German minesweepers escorting a merchant ship. Again, this battle was indecisive showing the difficulty of engaging in nighttime coastal operations with a veteran force like the 24th. The Channel Islands remained in German hands throughout the rest of the war and its naval forces could still get a few licks in such as when the subchaser PC-564 had to flee four minesweepers when its gun jammed and she had to be grounded at Pierre de Herpin 8 March 1945.

PC-564 and PT509 were not the only USN ships lost to German surface attack. One of the more serious battles in the English Channel was the S-boat attack on Exercise Tiger off of Slapton Sands in April 1944. Tiger was part of the training and rehearsal for Overlord and consisted of eight USN LSTs and a RN corvette for escort. The S-boats attacked the heavily-laden LSTs, sinking two- LST-507 and LST-531- and severely damaging two more. The cost of Slapton Sands was also quite severe with over seven hundred personnel lost and it caused minor concerns over security since some of the dead knew details of Overlord and it was not known if the Germans had picked som up as prisoners.

The S-boats also engaged with Overlord proper and the Cherbourg flotilla managed to sink a few USN LSTs and some other small craft. These attacks took place at night and were pinpricks to the larger Neptune fleet. Mines and obstacles were a greater threat to the Allied shipping in the region. German shore batteries, which were often manned by Kriegsmarine personnel, were responsible for the loss of the destroyer USS Corry off of Utah beach and the same battery likely sunk the patrol craft PC-1261, which some historians consider the first Allied warship lost to direct enemy action in Overlord.

Overall, the USN's contributions to the Allied naval effort in European coastal waters was minimal outside of merchant shipping. The USN had some heavy units involved in the main operations, but they supplemented the largely-RN forces which did the bulk of the heavy lifting. Vincent O'Hara estimates of the thirty-eight major engagements (in this case, O'Hara excludes small craft attacks like S-boats), the Americans were present in less than five percent of the engagements, coming behind the Soviet Navy. But while battles such as Imperia were not nearly on the scale of Leyte Gulf or the Philippine Sea battles, they were real enough for the men who fought them. The battle of Slapton Sands was by some metrics the last significant Axis naval victory of the war; the Kriegsmarine managed to interrupt and Allied training exercise and extract a high number of dead with no losses of their own. But Slapton Sands remained an exception rather than the rule for the Kriegsmarine's last year and most of their engagements were not as one-sided.