Beyond the submarine actions described in the comments linked by /u/DanKensington, there were sorties by some German naval units against the Operation Neptune forces.

In the early hours of the 6th June, German coastal observers and radars had been spotting large numbers of ships offshore. There were also reports of heavy bombings and paratroopers being dropped in the Normandy region. All of these reports led to considerable confusion at the headquarters of Naval Group West, the Kriegsmarine command responsible for the defence of the Atlantic coast. Naval forces were ordered to sortie from Cherbourg and Le Havre to find out what was going on. Two flotillas of S-boats were ordered out from Cherbourg, but had to turn back due to bad weather. Three German ships sortied from Le Havre at 4:45 AM. These were the torpedo boats (small destroyers) Mowe, Jaguar and T-28. Another seven ships of the 15th Vorpostenboot-Flottille were ordered to be made ready, though these were mostly armed trawlers.

The three torpedo boats encountered the Allied flotilla at 5:22, hitting the smokescreen the British had laid to cover the eastern flank of the landings. Breaking through the smokescreen, they engaged the Allied convoys heading to Sword Beach. They fired a spread of 18 torpedoes, and turned back through the smoke before the Allies could respond. Only one of these torpedoes scored a hit, on the Norwegian destroyer Svenner. The torpedo broke the back of the ship, and she sank quickly. There were a few near misses, including one on the HQ ship Largs, flagship of the Sword landings. On the 6th, three German destroyers, based in the Gironde were ordered north to Brest, to join up with another ship there.

In the days following the landings, there were a number of further skirmishes between the Allied forces and German light forces. During the night of the 7th June, a flotilla of S-boats from Cherbourg made an abortive attack on the Allied forces, as did the torpedo boats from Le Havre. At the same time, the German destroyers left Brest to make a run into the landing areas. The RN's 10th Destroyer Flotila, combining British, Polish and Canadian ships, would engage them off the coast of Brittany on the night of the 8th-9th. Two of the German destroyers would be sunk, and a third heavily damaged. The torpedo boats from Le Havre would make more attempted sorties. None of these would see any success, but the British counterattacks saw no success either. The S-boats were more successful. Operating from Cherbourg, and from bases further afield, they made night attacks on Allied convoys. Over the course of the first week after D-Day, they sank or heavily damaged five amphibious vessels, three merchants, a frigate, a tug, and a few coastal craft. Even then, this was a tiny fraction of the invasion fleet. In return, the S-boats took a number of losses to Allied mines, ships and aircraft. On the 14th June, RAF Bomber Command launched a massed raid against Le Havre. This sank all three of the torpedo boats, ten S-boats, an R-boat (coastal minesweeper and seventeen patrol boats. The port was rendered useless. The next night, Boulogne was attacked, to similarly devastating results. Meanwhile, Cherbourg was evacuated after American forces were able to close off the Cotentin Peninsula, cutting off the port. With this, the immediate threat to the landing beaches was removed.

The largest German units to attempt an attack on the beaches were the destroyers engaged by the 10th Destroyer Flotilla on the 9th June. None of the Kriegsmarine's cruisers or battleships attempted to attack. This was partly because earlier Allied attacks had rendered them useless. Tirpitz had been heavily damaged by RN midget submarines, then re-attacked and damaged by aircraft from RN carriers. Gneisenau had been put out of action by RAF bombers. The other main aspect was German actions. Hitler had been angered by the failure of German surface forces to destroy a weakly escorted convoy during the Battle of the Barents Sea. As a result, he would (after some persuasion by Donitz and Raeder) order the German fleet's heavy units to remain in port to act as a fleet in being. This was only confirmed after the loss of Scharnhorst at the Battle of North Cape. As a result, even if Tirpitz or Gneisenau had been available, they would not have been used. A lack of fuel, and to a lesser extent, spare parts, also helped confine them to port.

However, the Allies were not aware of this. They were worried that the German fleet could sortie and attack the landing force. The plan to combat such an attack was named 'Operation Hermetic', and called for air attacks by land-based Coastal Command and Fleet Air Arm units before an attack by the cruisers and battleships of the Eastern Bombardment Task Force. However, it was considered that the Germans might not attack the invasion fleet. Instead, they could attempt to use the confusion to break out into the North Atlantic, to attack Britain's lifeline of convoys. To counteract such a possibility, the Royal Navy maintained a considerable strength at its main base in Scapa Flow.

There wasn't much of a fleet left to sortie with. They did try with their U-boats, as u/thefourthmaninaboat and u/wotan_weevil examine in their answers in this thread.