American Historians always emphasize the importance of the USS Monitor and CSS Merrimac in Ironclad naval warfare citing it as the first example of ironclad on ironclad fighting. When did Europeans adopt this style? Are there any notable battles between Ironclad ships in the years before World War I?
They were in active commission before the ACW began. Both the French Glorie class and the HMS Warrior and her sisters were built and designed around 1859.
Both ships were also full frigates mounting up to 40 guns, most of them breechloader rifles and of greater range and weight then the Dahlgren smoothbores in the USN though the Dahlgrens were an excellent advancement of smoothbores and remarkable in their safety record.
Both Britain and France had experience with armor plated floating barges in the Crimea.
So yes Britain and France had large ocean going ironclad warships even before the war. Warrior is preserved as a museum ship and a search for her picture (sorry im on mobile) should give an idea just how much bigger then even the Virginia we are talking.
The French came up with the notion of attaching iron plates to floating batteries during the Crimean War, as a way of defending against the explosive shells fired from Paixhans guns. French Dévastation-class ironclad floating batteries would prove very useful at reducing Russian forts during the Battle of Kinburn in 1855.
Following those successes, the French Navy decided to scrap most of their planned construction of conventional wooden steamships, and instead pursued the goal of ocean-going ironclad warships. The Dévastation-class batteries were only capable of four knots and would not have been capable of sailing in rough waters, so something better was clearly needed. In 1858, construction began on the ocean-going ironclad Gloire, which was launched the following year.
I talked a bit more about those batteries in this past thread.
The French and British had shared information during the Crimean War, and the British too had built their own ironclad batteries. In 1860, the British launched HMS Warrior, which surpassed even Gloire.
Given that Gloire and Warrior were completed before USS Monitor and CSS Virginia (USS Merrimack was the name of the ship that the Confederates turned into their ironclad Virginia), one can say that the revolutionary reputation of Monitor and Virginia is perhaps exaggerated at times. They are certainly more radical looking than their French and British counterparts, a result of being built as brown-water ships which didn't have to deal with rough seas.
As for battles, the Battle of Lissa in 1866 between Italy and Austria saw both sides use ironclads in addition to regular wooden warships (the Italians had 12 ironclads, the Austrians 7). The Austrians, under the leadership of the very capable Wilhelm von Tegetthoff, prevailed. One of the interesting things to result from Lissa was the rise in the popularity of ramming as a theoretical tactic. The Austrians pulled off a few ramming attacks on Italian ships in the battle. However, ramming didn't really prove to be a promising lead in the development of naval warfare.
The Battle of Hakodate in 1869, a part of the Boshin War - the civil war between Imperialist and Shogunate forces in Japan - featured the use on the part of the Imperialist forces of the ironclad warship Kōtetsu, a ship that was actually initially built by the French for the Confederate Navy. The rest of the ships at that battle were not ironclads, however.
The Battle of Angamos between the Chilean and Peruvian Navies in 1879, a part of the War of the Pacific, featured a few ironclads. The Chileans prevailed. The Peruvian ironclad Huáscar was captured, and is now a museum ship in Talcahuano, Chile.
As ships built fully out of iron and steel (rather than being clad in it) became the norm, the term "ironclad" fell out of use. There were wars fought before WWI that featured battles between pre-dreadnought battleships, cruisers, and the like. You might have heard of the famously lopsided Battle of Manila Bay during the Spanish-American War, in which the US sunk seven Spanish ships whilst suffering almost no losses. The Imperial Japanese Navy was pretty active in the late 19th and early 20th century as well. They defeated the Chinese Beiyang Fleet (which was well-equipped with European-built warships) at the Battle of Yalu River in 1894, a part of the First Sino-Japanese War. They also famously triumphed over the Russians during the Battle of Tsushima in 1905, a part of the Russo-Japanese War.
Source: Baxster, James P. The Introduction of the Ironclad Warship. Naval Institute Press, 1933.